OK. Let's try again. This is an example about the importance of background probabilities--the importance of taking statistics seriously--that's been used by every economist I've talked to who has ever taught in a law school. (And they almost invariably say that their students do not get it.)

A bank robber puts the loot in a suitcase, runs out of the bank, gets into a taxi, and is driven off. A witness sees him. The witness says, "It was a black taxi, not a yellow taxi." Further tests reveal that this witness get's the color of the taxi--whether it is black or yellow--right 80% of the time. So the police dispatcher radios all cars, saying "He's in a taxi. There's an 80% chance it's a black taxi."

Now a supervisor hands the dispatcher a slip of paper that tells the dispatcher that 90% of the taxis in the city are yellow, and only 10% are black. "It's unlikely that the taxi is black," the supervisor says. The dispatcher begins, "Correction..." What should the dispatcher say?

A *law professor* says: "The dispatcher should say, 'No correction.' The taxi is either black or yellow. We have a witness. We know the reliability of the witness. This is a matter of fact, and statistics do not matter." the law professor is wrong.

A *statistics professor* says: "Let's back up the probability tree. The bank robber runs out of the bank and sees a taxi. Suppose this happens 100 times. 90 times the robber gets into a yellow taxi. 10 times the robber gets into a black taxi. The witness then sees the taxi. Out of our 100 times, 72 times the robber will get into a yellow taxi and the witness will say the taxi was yellow, 18 times the robber will get into a yellow taxi and the witness will say the taxi was black, 8 times the robber will get into a black taxi and the witness will say the taxi was black, and 2 times the robber will get into a black taxi and the witness says the taxi was yellow. We know that the (72) and the (2) chances did not happen--the witness said the taxi was black.

"So we know we are either in a state of the world that happens 18 times out of a hundred--the robber gets into a yellow taxi and the witness says the taxi was black--or a state of the world that happens 8 times out of a hundred--the robber gets into a black taxi and the witness says the taxi was black. That means that the odds are 18 out of 26--69%--that the taxi is yellow, and only 8 out of 26--31%--that the taxi is black.

"The dispatcher should say, 'Correction: there is a 69% chance that the taxi is yellow and only a 31% chance that the taxi is black.'"

The statistics professor is correct.

The Bennett case is analogous. Bennett is the witness reporting that he is "nearly even." But this time the background odds are not 90%-10%, but rather--by the law of large numbers--greater than 99.99% that Bennett has lost a fortune, and less than 0.01% that he is "nearly even."

Posted by DeLong at May 7, 2003 11:54 AM | TrackBack

Comments

While I'm no statistician, and I'm sure the statistical analysis is correct, I don't see that we have any choice but to go the law professor's route. Let's change the example by leaving out the one thing we never know about witnesses, their rate of reliable identifications of specific phenomena, and make it somewhat more realistic.

A is run over by a bus coming out of the Port Authority bus station at a time during the night when several different buses were leaving. W saw the bus and says it's a Trailways bus. A sues Trailways for negligence. We know nothing about W's general ability to recognize buses at night. Assume 70% of the buses at the Port Authority are Greyhound, 7% are Trailways, and 23% "other," and that nothing about the particular time of the accident would change the percentages.

W testifies that the bus was a Trailways bus. Trailways, which has impleaded Greyhound, puts up an elaborate and probably correct statistical analysis showing a high probability that it was a Greyhound bus, despite what the witness said.

As I see it, the statistical analysis is fair game for impeaching W, making it seem less likely that his perception was accurate, but I can't see how we can fairly use it as a basis for finding Greyhound liable. To the extent that it's valid, it's valid whether there was a witness or not. Simply because 70% of the buses are Greyhounds, there's an X% probability that the negilgent bus driver was a Greyhound driver, a much greater probability than that any other bus company is at fault. Therefore, why bother with a witness at all? Why not market share enterprise liability based on the respective bus companies' share of the market? What if Greyhound has 70% of the market precisely because it has better drivers?

A was run over by whatever bus company actually ran him over, not by a percentage of one bus company and a percentage of another. Long shots come in. Unless we want to scrap the system of trying to determine as best we can what actually happened and getting compensation from the responsible party with a system of social insurance based on what, actuarially, would be expected to happen, do we have any choice but to go with the witness over the statistician?

A wrinkle in the oinment, possibly: one probability missing is the probability of a black taxi of coming by the particular bank at that particular time. For instance, it may be that the company(s) that have black taxis could be heavier in the bank's area then yellow taxis. In other words, one must throw in the assumption that all taxis from all companies have an equal chance to come by that bank at that time.

If I am right, this just shows that the lawyer may be right, but Bennet is still wrong. His chances of being "nearly even" are less likely to have mitigating circumstances then the taxi passing a bank. For example. it is much less likely that a slot machine was tinkered with so that its odds were as good as Bennett needs to "break even". It is much moire likely that something like unevenly distributed black taxis in the bank's neighborhood to be true.

Ben Ricker

Posted by: Ben Ricker on May 7, 2003 01:00 PMCJ,

Interesting point, but I'll have to disagree. At what point would you be willing to stop implicating Trailways? When there's one bus in a thousand? In a million? Obviously, at some point you start realizing that this is a witness who is not terribly good at recognizing bus brands.

So the Right Thing to Do (says the psychologist, pretending to be a statitician) is presumably to show the witness a few hundred movie clips of buses going by and find out their reliability. From there, you have a prior probability.

If you don't do something like that, or use some set of well-known figures, you have to assume *some* prior probability of the witness' accuracy--but 100% ("we have no choice", you wrote) seems like an odd choice.

If there had been no witness, after all, we still would have had some sort of case. So having an unreliable witness gives us more information, but not complete information.

Posted by: Dan on May 7, 2003 01:04 PMA wrinkle in the oinment, possibly: one probability missing is the probability of a black taxi of coming by the particular bank at that particular time. For instance, it may be that the company(s) that have black taxis could be heavier in the bank's area then yellow taxis. In other words, one must throw in the assumption that all taxis from all companies have an equal chance to come by that bank at that time.

If I am right, this just shows that the lawyer may be right, but Bennet is still wrong. His chances of being "nearly even" are less likely to have mitigating circumstances then the taxi passing a bank. For example. it is much less likely that a slot machine was tinkered with so that its odds were as good as Bennett needs to "break even". It is much moire likely that something like unevenly distributed black taxis in the bank's neighborhood to be true.

Ben Ricker

Posted by: Ben Ricker on May 7, 2003 01:05 PMA wrinkle in the oinment, possibly: one probability missing is the probability of a black taxi of coming by the particular bank at that particular time. For instance, it may be that the company(s) that have black taxis could be heavier in the bank's area then yellow taxis. In other words, one must throw in the assumption that all taxis from all companies have an equal chance to come by that bank at that time.

If I am right, this just shows that the lawyer may be right, but Bennet is still wrong. His chances of being "nearly even" are less likely to have mitigating circumstances then the taxi passing a bank. For example. it is much less likely that a slot machine was tinkered with so that its odds were as good as Bennett needs to "break even". It is much moire likely that something like unevenly distributed black taxis in the bank's neighborhood to be true.

Ben Ricker

Posted by: Ben Ricker on May 7, 2003 01:08 PMCJ,

Interesting point, but I'll have to disagree. At what point would you be willing to stop implicating Trailways? When there's one bus in a thousand? In a million? Obviously, at some point you start realizing that this is a witness who is not terribly good at recognizing bus brands.

So the Right Thing to Do (says the psychologist, pretending to be a statitician) is presumably to show the witness a few hundred movie clips of buses going by and find out their reliability. From there, you have a prior probability.

If you don't do something like that, or use some set of well-known figures, you have to assume *some* prior probability of the witness' accuracy--but 100% ("we have no choice", you wrote) seems like an odd choice.

If there had been no witness, after all, we still would have had some sort of case. So having an unreliable witness gives us more information, but not complete information.

Posted by: Dan on May 7, 2003 01:09 PMNothing makes my eyes droop quite like stats. Even my econ books weren't so painful :).

Posted by: Stan on May 7, 2003 01:19 PMBrad (and others), aren't you getting a little carried away by this 'extremely unlikely' thingy?

The problem is in analyzing extreme payoff situations as though they were coin flips.

Consider:

me: I play the lottery every day all my life and I broke even

Brad: That is impossible. You are 40 years old and have dropped $14,600 lifetime on lotteries (dollar a day all your life). With a standard deviation of about $1000, yours is a 15 SD event - probability is so remote that is is not even worth computing.

me: I won 40 million yesterday. Check the papers. here is my picture with the mayor.

Brad: But...

Posted by: Suresh Krishnamoorthy on May 7, 2003 01:21 PMBut is Bennett's gambling an extreme payoff situation? I think not. Certainly there are no extreme payoffs in video poker (although good hands can pay off sizably). As far as slot machines go, I don't know. But I don't think we're talking about lottery sized winnings in that case, either. Besides, if he had hit the big slots payout, wouldn't that have come out. "Were it not for Bennett winning a $5 million, he'd be way under ..." or something along those lines.

Posted by: Rafe on May 7, 2003 01:25 PMSuresh, you're ignoring Brad's point about witness accuracy.

Here's the thing: lots of gamblers say they break even. In practice, very few do. (My friend Terence informs me that when you hear a gambler say that he breaks even that means he's broke). So, the probability that a randomly chosen gambler breaks even is very low. The conditional probability that a gambler who says he breaks even actually does is somewhat higher but still very low.

The Statistics professor is wrong. I ask that Dr. DeLong provide evidence showing that any audit of $5 or greater slot machines in Las Vegas during the time period at issue have even come remotely close to a 90% payout. There may be some as low as 96%, but I would venture that most are much better than that. Additionally, it is very likely that the payoffs on $500 slots approach 99% or higher. Your statistical analysis is sound, but one of your key factual assumptions is wildly off the mark unless Bill Bennett lost his $8 million playing nickel slots at the airport where he may have been only getting an 88-93% payout.

Posted by: Kevin H on May 7, 2003 01:31 PMRafe,

I don't disagree with you. You are correct - the story would have been played that way.

Personally I think that Bennett is bluffing, but *not* for the reasons of statistics.

I maintain it is incorrect to use statistics and the CLT in situations where probability distribution functions collapse (like Schrodinger's Cat)

To paraphrase a famous novel I once read: 'What are the odds that four planes would be simultaneously hijacked and flown into two buildings and totally destroying them? On paper they don't exist.'

In Bennet's case, we may well be working with an event *after the fact*, where all the distributions in the world have no relevance.

Brad's argument holds *only* if Bennett came to him and said, 'I intend to drop $8 million on slots over the next few years and I expect to break even'.

Posted by: Suresh Krishnamoorthy on May 7, 2003 01:34 PMTwo comments:1. Bayesian thinking is important in medicine too. 1. If the pre-test likelihood of a disease is very low, a positive result of even a highly specific test will likely be a false positive. 2. Sure the stat professor is right, but (God luv 'im) he still looks, acts, and talks like a goofy math major-variant. The law prof, in contrast, will probably have a good haircut, shined shoes, and a mellifluous voice. Which would look better in front of a jury of daytime-TV watchers?

Posted by: j rossi on May 7, 2003 01:48 PMTwo comments:1. Bayesian thinking is important in medicine too. 1. If the pre-test likelihood of a disease is very low, a positive result of even a highly specific test will likely be a false positive. 2. Sure the stat professor is right, but (God luv 'im) he still looks, acts, and talks like a goofy math major-variant. The law prof, in contrast, will probably have a good haircut, shined shoes, and a mellifluous voice. Which would look better in front of a jury of daytime-TV watchers?

Posted by: j rossi on May 7, 2003 01:50 PMEric,

Here's the thing: lots of gamblers say they break even. In practice, very few do.

So, we are now really calculating the probability 'Bennett is lying' which is very different from 'No one can possibly spend $8 MM on slots and not lose a ton of money'

See my post above about *events that have already happened* but are unknown to the observer.

Probabilities applied to events that have already happened are not probabilities of the event happening. They are probabilities of someone guessing the event correctly.

Consider: I drew a glove from a box of gloves. What is the probability I hold a right hand glove?

should really be:

if you guessed that I hold a right hand glove in my hand, what is the probability that you are correct?

This is how an event that has happened can be properly analyzed with a probability - by transforming it into an event that is about to happen.

Posted by: Suresh krishnamoorthy on May 7, 2003 01:52 PMI won't try to improve on Brad's explanation, but I do have to say, the flat refusal of seemingly intelligent people to accept statistical arguments is pretty amazing.

Suresh, are you really saying that in choosing between two possible past events both of which are consistent with the evidence at hand, we have to ignore probability altogether?

Wait a sec… something seems wrong—your variance 1350 makes sense—but it so dominates the mean that it should be easily possible for Bennett break even. Isn’t it true that the very high variance cancels out the 10% bias?

Let’s assume that you are right, and Bennett plays slots that have a payback percentage of 90%. [This helps us ignore the actual structure of payouts from the machine.] Over time, out of every dollar that Bennett bets, 10% goes to the house. As you point out, to lose $8 million on slots, Bennett, on average would have bet $80 million, and played 160,000 times. If he were to go to Vegas 10 times a year for 10 years, this would mean he had to play 1,600 games per visit—OK. Given that it takes 15 seconds per game, that means 4 games a minute, or 240 an hour—or between 6 and 7 hours. That’s OK.

However, that is too simplistic—after all statistics are very important. If we use the law of large numbers and central limit theorem, we can use a normal distribution to approximate a series of independent pulls at a $500 slot. Using your assumptions that the mean loss of one pull is (10%)*500=50 and the standard deviation is 1350, then for 160,000 pulls, the mean loss is (10%)*500*160,000 or 8,000,000 and the standard deviation is 160,000*1350=175,000,000. The probability of losing only zero (or gaining) is approximated by the P(Z>(0-8)/175)=1-P(Z<=.05)=1-.5199=48%

Am I calculating something wrong? [It has been a long day]

It's very tricky to use Bayesian updating with witness testimony. Bennett is almost certainly lying, but you need to know that he is a politician and has a motive to lie to figure that out.

Suppose Bennett is applying for a job, his prospective employer asks for his SSN and phone number, and he provides it. A priori, the odds of someone having those two numbers is something like 1 in 10 quadrillion. Yet, such is the power of (unbiased and unprejudiced) testimony that he probably provided the right numbers.

It's only once you know that Bennett has a reason to lie about the gambling winnings that the a priori unlikelihood of his coming out even on the slots becomes dispositive. There isn't a universal witness reliability number that can be plugged in to all cases.

Posted by: Mike on May 7, 2003 02:15 PMIt's very tricky to use Bayesian updating with witness testimony. Bennett is almost certainly lying, but you need to know that he is a politician and has a motive to lie to figure that out.

Suppose Bennett is applying for a job, his prospective employer asks for his SSN and phone number, and he provides it. A priori, the odds of someone having those two numbers is something like 1 in 10 quadrillion. Yet, such is the power of (unbiased and unprejudiced) testimony that he probably provided the right numbers.

It's only once you know that Bennett has a reason to lie about the gambling winnings that the a priori unlikelihood of his coming out even on the slots becomes dispositive. There isn't a universal witness reliability number that can be plugged in to all cases.

Posted by: Mike on May 7, 2003 02:16 PMI think the problem with the cab example is that most people believe a witness would know the color of the cab at a much higer percentage than 4 out of 5. Our intuition works against us, because the hypo is so unrealistic.

Posted by: markmeyer on May 7, 2003 02:17 PM>>the standard deviation is 160,000*1350=175,000,000<<

You need to take the square root of 160,000..

Posted by: Brad DeLong on May 7, 2003 02:26 PMYeah, I see it. Thanks.

Posted by: Kevin Brancato on May 7, 2003 02:28 PMLet's see:

Casino sources say that Bennett had credit lines of up to $200,000.

Those same sources say that Bennett lost around $8 million over the last few years.

As DeLong says, that would mean, assuming payouts of 98%, that Bennett gambled $400 million over the last few years.

On credit lines of up to $200,000.

Gambling on occasion, for a few hours at a time.

Now, what do we know?

1. It is difficult to imagine anyone beating the payout % over a period of time.

2. It is very unlikely that Bennett in fact beat the payout % over the last few years.

3. Bennett very likely lost money.

4. For Bennett to have gambled $400 million, he'd have to have been carrying some very big suitcases.

I'm sticking with Volokh. Sometimes the argument isn't about statistics. In this case, it's about the fact that DeLong has seized on the $8 million figure despite its improbability.

Posted by: Thomas on May 7, 2003 02:34 PMKevin B.

According to the CLT, the variance of the limit distribution is 1350^2/165,000. So the s.d. is approximately 3.

Posted by: dlapple on May 7, 2003 02:34 PMI know, incidentally, that eye-witness identification of strangers, even unbiased and unprejudiced strangers, is very unreliable. My point is that the reliability of people varies so radically in different circumstances that one can't just point to background odds and be done with it.

Posted by: Mike on May 7, 2003 02:37 PMI believe I am absolutely incapable of learning statistics. Ok, maybe not absolutely: I guess if I were stranded on a desert island with nothing but a statistics textbook, I might pick up some of the basics.

In terms of the Bennett case, I have an idea (an unstatistical one) that it is extremely unlikely he could have broken even after betting so much money over so many years. This stems from my assumption that if it were anything less than extremely unlikely that someone could gamble like that over the course of a decade and not lose a lot of money in the process, then it wouldn't be profitable to run casinos and Vegas wouldn't be the den of iniquity and high temple of tackiness that it so clearly is. But if someone suggests that I should be "agnostic" on the question, or even that I should rather assume it is moderately to very likely that he broke even, I am at a loss to respond.

My question (and it's a statistical one of sorts): given this taxicab example, what are the odds that a randomly selected group of ten duly certified statistics professors would come up with significantly different answers? In other words, can't I just trust the nearest statistics professor I can find for expert knowledge and leave it at that?

Of course this doesn't really solve the problem of statistical illiteracy (er, innumeracy). We are inundated with statistical information and misinformation which we don't know how to interpret, and we don't generally refer the issues to statistics professors.

Posted by: Invisible Adjunct on May 7, 2003 02:39 PMDan:

Maybe I should have been clearer. I don't suggest that a jury HAS to believe the witness and hold Trailways responsible, and I think it fair game to use statistics to show that he may be wrong. If the statistical likelihood of his being right approaches lottery odds, then a judge should be able to take the case against Trailways away from the jury and dismiss it.

My problem is with allowing the jury, with nothing more than a witness who says "Trailways" and statistics that say "Greyhound," to decide against Greyhound. The statistics that say "Greyhound" will say "Greyhound" whether there is a witness or not and really say nothing more than that even before an accident happens we can make a smart bet that the bus will be a Greyhound. If we can do that, we can eliminate the "middleman" of a trial at all. Whenever there's a bus accident, assess each bus company for its proportional sghare of the damages based on market share. That's a form of social insurance that may, wholesale, be more just and efficient than trying to pin all the damages on the particular bus company responsible retail, and there may be a lot to be said for it. But as long as were are doing what we are doing instead, I don't see any alternative to insisting on witnesses rather than statisticians.

Suppose I were to engage Shaquille O'Neal in a casual game of 1-on-1. Suresh's logic suggests that, although going into the game there would be astronomical odds against me winning, after the fact those odds would have no bearing on the question of whether it's credible for me to claim I'd beaten Shaq.

This is obviously ludicrous. No matter what I tell you after the fact, it is very, very unlikely that Shaq didn't CRUSH me. If you choose to believe me when I say I did, you are choosing to go AGAINST the evidence.

However: in terms of the nuts and bolts calculations involved, Suresh's reasoning does touch on a flaw in frequentist statistical thinking - when discussing past events, it is cumbersome to think of "probabilities" as anything other than measurements of PERSONAL UNCERTAINTY (as opposed to numbers based in nature), a problem that Bayesians such as LJ Savage have articulated to a point where (IMHO) it is established beyond argument (and I'm a frequentist, most of the time). In other words: either Shaq beat me, or he didn't; the after-the-fact "probability" that he beat me is either 1 or 0. It's your uncertainty about this thing which can be described by "probability," and that uncertainty is informed by prior knowledge.

But: objectively speaking, the evidence STILL very, very strongly suggests that I got beat - precisely because the probability that I would get beat, GOING IN to the game, was so high. We could get into rigorous examination of how the calculations quantifying the degree to which the data are suggestive should be performed (personally, I'd use a likelihood curve, but not all statisticians agree).

But the overall argument is one of common sense: if I claimed to have beaten Shaq, you'd laugh.

So: no matter how you view the details, the data are NOT 'agnostic' about whether Bennett is lying. If an observer like Volokh decides HE is 'agnostic,' that's because he's worse than ignoring the data - he's going against it in a big way, based (presumably) on Bennett's word.

He's obviously a clear thinker, making a simple mistake. There is just no way he'd say he's agnostic on whether I just wrecked Shaq in basketball, no matter how much I insisted I had.

Posted by: ryan on May 7, 2003 02:55 PMPersonally, I don't believe for a moment that Bennett played the slots regulary for ten years without paying the house very close to its full due.

OTOH, in real life court cases there's typically all kinds of other evidence that statisical evidence and assertions should be considered in conjunction with.

Take physical evidence: If Bennett had really pulled slot machine levers enough times to bet $400 million or so, wouldn't he either be suffering from Repetitive Strain Injury or be boasting one hell of a developed forearm?

Absent such evidence, I take it that the statistical law implies that the statistical assertion from "some" anonymous sources is exaggerated.

(Few anonymous sources understate assertions for the sake of a good story. Then again, I haven't seen his forearm.)

Re: Wagering $400,000,000

Not hard to believe at all. The total you wager is not the same as the total of all the money you ever took to the casino.

For example:

Assume I take $10,000 to the casino and play slots at $250/pull and I pull the lever, on average, 40 times per hour.

During the first hour I'll wager $10,000. If the payout is 98%, then my expected stake remaining at the end of the hour is $9,800. I've wagered $10,000 and still have most of my stake left. If I keep going until I've lost it all, it will take several hours and I'll have wagered much, much more than the $10,000 I brought with me.

Posted by: Chip on May 7, 2003 03:38 PMSuresh,

While your point about selection bias in the Bennett situation is valid, you point about 9/11 is not. We have absolutely no way of estimating the probability of events such as the simultaneous hijacking of four planes. Not only are they not independent events, but they are dependent on so many different variables that any kind of calculation is made meaningless. If you want a much more thorough treatment of risk vs. uncertainty, I would refer you Nassim Nicholas Taleb’s treatment of the subject in a paper titled Random Processes, Opacity, and Knowledge: The Central Problem, which can be viewed on his homepage (just search for his name on Google). He also has an excellent book titled Fooled by Randomness: The Hidden Role of Chance in Markets and Life.

However, back to Bennett… Although it is impossible to compute the “true” odds that he actually has come close to breaking even (as we don’t know what his propensity for lying is, or the propensity of the casinos that probably leaked the information, or what his subjective judgment of “even” amounts to), I sure wouldn’t bet my money that he has.

Nathaniel

ryan,

your example is the logically identical to the glove example i gave, except the odds are much more skewed.

there are *two* distinct events here:

1: The odds of you beating shaq on a 1-on-1

2: The odds of you *having beaten* shaq on a 1-on-1

Once the game has been played, (1) is meaningless.

(2) is subject to analysis, BUT, it has nothing to do with (1).

I have to be particularly foolish to bet in favor of you beating Shaq,

*but that does not make it impossible for you to have beaten him - no matter what the odds originally were, as long as they were non-zero*

-he could have had a bad day, you might be his favorite nephew and he lost on purpose, Harry Potter cast a spell on the ball -

So, the probability that Brad calculates is answering the question: If i bet that Bennett did not break even, what is the probability that I am right? Once we see Bennett's tax returns from the IRS, that function also collapses.

This is the Schrodinger's cat paradox that led to his equations of subatomic particles being in a probability cloud.

Posted by: suresh krishnamoorthy on May 7, 2003 03:39 PMJim Glass says (paraphrased): wouldn't Bennet have hurt himself, pulling a slot lever so many times?

FYI: Most modern slot machines don't have levers, you just push a button to make them go. Most casinos are populated almost entirely with the push-button variety, although some retain a few old lever-operated types to provide flavor. But, the push-button types are preferred because they allow for faster (and thus more) transactions, and people can sit in front of them for hours without their arms getting tired and requiring a break.

What if the standard deviation of the slot payouts is really much higher than 1350? I found a page (http://krigman.casinocitytimes.com/articles/5374.html) that reports sample payout schedules, and I calculated the standard deviation of a 90% payback machine (Machine D in the bottom right) as 7100. Everything else the same, this yields a Z of 2.8, and a miniscule probability of Bennett breaking even of 0.24%.

However, if Bennett used machines of 95% payback, then we get a different answer. And that answer all depends on the variance of the payoffs. If I modify the highest payoff in the table from 0.003% to 0.005%, Machine D become a 95% payback machine, and the standard deviation rises to 9036. This gives a Z of 2.2 and a much better ,though still very small, chance of breaking even of 1.4%.

Anybody else try a few variations?

"I ask that Dr. DeLong provide evidence showing that any audit of $5 or greater slot machines in Las Vegas during the time period at issue have even come remotely close to a 90% payout. There may be some as low as 96%, but I would venture that most are much better than that. Additionally, it is very likely that the payoffs on $500 slots approach 99% or higher. Your statistical analysis is sound, but one of your key factual assumptions is wildly off the mark unless Bill Bennett lost his $8 million playing nickel slots at the airport where he may have been only getting an 88-93% payout."

Yes. Bennett almost certainly lost money, but the keys to how much he lost are: 1) the total amount he bet on each type of machine, and 2)the percentage loss for each of those machines. Here's an opinion piece by someone who appears to know the specific percentages:

http://slate.msn.com/id/2082638/

That piece estimates the payout on low-stakes machines at 90 to 95 percent, and high stakes machines at 98 percent. And it therefore estimates his losses at 2-10%.

Using these 2-10% figures, if he lost $8 million, he would have put in somewhere between $80 million and $400 million. Figuring he makes even $10 million a year ($200,000 a week, i.e., $50,000 a lecture, 4 times a week), it would be awfully hard to imagine him putting in $80 to $400 million, over a decade. In fact, it would be flat out impossible for him to put in $400 million, as he would "only"--ho, ho, ho! that's kind of funny for me to write--have earned $100 million for the decade.

Posted by: Mark Bahner on May 7, 2003 03:45 PMoops:

i meant (2) is very different from (1) - since (1) has now been determined.

Posted by: Suresh Krishnamoorthy on May 7, 2003 03:50 PMIn the taxi case, I agree with markmeyer. I basically just threw out the "80% accurate identification" as junk science. It seems unlikely to me that observers would mistake yellow for black or black for yellow 20% of the time, especially if the "don't know" option is allowed. This is probably a problem for anyone writing statistical "story problems" -- making sure that the story convincingly fits its statistical description.

As to how much Bennett lost, net, that's up in the air. $8 mil at 2% would take a lot of hours on a lot of days.

Knowing what proportion of the money is tied up in widely-spced enormous pots still seems relevant. For one thing, if Bennett had won a million-dollar or ten-million dollar pot we probably would have heard about it.

Posted by: zizka on May 7, 2003 04:04 PMSuresh, you wrote that there are two events:

"1: The odds of you beating shaq on a 1-on-1

2: The odds of you *having beaten* shaq on a 1-on-1

Once the game has been played, (1) is meaningless."

(1) has great bearing on what should be our imperfect assessment of what likely DID happen, even after the fact - on whether I "have beaten" Shaq, in the absence of other knowledge. The fact that the game has already occurred does not invalidate the relationship between the two, if the "odds" in (2) are defined in terms of personal uncertainty.

In other words, you obviously can't know the truth without some sort of proof, but you can objectively and factually state what the data say about the possibilities, and the degree to which they say it, and that argument is informed by the probability law going into the game.

Denying this is an assertion that statistical analyses are meaningless. If I develop lung cancer, is medical science retrospectively 'agnostic' on whether 80 pack-years of smoking is likely to have had something do to with my predicament?

Of course not. While it's true that either my lung cancer was caused by smoking or it wasn't, we can quantify the degree to which the data themselves SUGGEST that smoking caused it. Then, whether we BELIEVE the data or not is a leap of faith - the degree of "leaping" we're doing is informed by the strength of the data and quality of the biomedical argument underlying it.

You say: "[Before the game,] I have to be particularly foolish to bet in favor of you beating Shaq." Just so, it would also be foolish to take my word for it after the fact. The data, as it currently stands, very strongly suggests that I'm lying (or, to be charitable in Bennett's case, profoundly mistaken*).

As you point out - if we could see Bennett's tax docs, then we could establish objective truth, and then talk of 'probability' would be meaningless - but in the absence of such truth, we are laboring under 'probability' as defined as personal uncertainty, and the data are much more suggestive of one possible truth than the other.

*Note that I'm not arguing the $8M figure here, but rather his assertion that he's "broken even." I've got no problem believing that there's someone out there who HAS broken even over that length of time; I just doubt that person happens to be Bennett.

>There may be some as low as 96%, but I >would venture that most are much better than >that. Additionally, it is very likely that the >payoffs on $500 slots approach 99% or >higher.

This is a classic example of a gamblers fallacy. What if you were to play a slot that offered an $8 million payout , cost $1 to play, and had a payoff of 100 percent. The answer is that 1 in 8 million times would there be a payout. When you see these 92 percent, 99 percent payouts, then the odds are even higher. The key thing here to understand is each trial is independent. The random number generator is programed in such a way that mathematically once in 8 million INDEPENDENT trials the numbers will hit. Bill Bennet, and others who lose large amounts don't understand this. They believe (falsely) that the trials are related in some way, so that a machine that has been played 7,999,999 times, is due to payout shortly. This how Bill Bennet could lose easily lose $8 million over a long period of time. His individual chances, determined by the random number generator, are low for each indpendent button push. He believes however that the more he plays the greater his chance of winning. This is fortified by his occasional wins. He puts his winnings back in and continues on.Even with a machine that pays out a 99 percent payoff any individual player will continue to lose far more than $1 on 100 repeated plays.

The postings here seem to reflect that the whole basis for the gambling industry is alive and well--ignorance of what a random number really is. (I am a consultant for Casinos).

Or, as Captain Spaulding didm't say, "Who are you going to believe, me or the evidence of your own calculations?"

Posted by: jam on May 7, 2003 05:09 PMThe recent revelation in no way diminish my longstanding admiration for William Bennett's ideas and writings, but they certainly tarnish my view of the man himself. And my primary reason is his BS assertion that he's come out "about even". This is about as plausible as Hillary's assertions about her options trading.

Bennett is not a hypocrite for gambling - he has never hidden it nor preached against it. Honesty, on the other hand, is at the core of his "Book of Virtues".

Lawrence -- Would I be correct in assuming that most $500 slots present the option to "play 1 to 5 tokens", and that the best % return comes with 5-token play? So the highest of high rollers would cycle $2500 (and at 98% return, lose $50) on each play?

Any statistics on "pulls" (plays) per hour (for single player continued play) on this class of machine?

Posted by: RonK, Seattle on May 7, 2003 05:19 PMMark Bahner wrote

>Using these 2-10% figures, if he lost $8

> million, he would have put in somewhere between

>$80 million and $400 million....In fact, it

>would be flat out impossible for him to put in

>$400 million, as he would "only"--ho, ho, ho!

>that's kind of funny for me to write--have

>earned $100 million for the decade.

You don't need $400 million on hand at any one time to wager $400 million over a decade -- unless you're making a single bet of $400 million. If you're making smaller bets, the odds are high that you'll get some money back, which you can use for later bets.

Posted by: MikeL on May 7, 2003 05:22 PM

Baysian statistics is all a bunch of elephant shit, as is the central limit theorem.

Score one for the de Long truth squad!

You're forgetting one other key piece of evidence: "The documents show that in one two-month period, Bennett wired more than $1.4 million to cover losses."

This is not proof that Bennett lost $8 million over the entire period, of course, but it is indicative of the ability to lose a great deal of money in a very short period of time, something that some of those above claimed was unlikely. Moreover, these kinds of losses would require some very large payouts to offset.

Personally, I think it highly improbable that Bennett came out "pretty close to even."

I think your defense of the law of large numbers and statiscal reasoning is too shrill and partisan, so I am going to disregard everything you say on the subject.

Posted by: roublen vesseau on May 7, 2003 06:45 PMIsn't the question what "pretty close to even" means? Statistical analysis can establish that if Bennett's done a lot of gambling, he's almost certainly lost money, but statistics can't tell you anything about how much he lost unless you know how much he bet. IIRC, Volokh said (1) the article says Bennett lost $8 million, (2) it's not clear whether the $8 million is net losses or something else, such as gross amount bet, (3) Bennett says he more or less broke even, and (4) I, Volokh, dunno.

If Bennett had bet a total of $8 million, he might have lost an amount that a guy in his position (rich person with gambling problem) could see as insignificant enough to assert that he had almost broken even ($250,000? $500,000). I suppose it's also possible that he lost a total of $8 million (net) in Casinos A, B, and C, but won enough in Casino D to feel like he was close to even. Statistics can tell us that Bennett is engaged in some degree of deception, but they can't tell us that he lost $8 million, because we don't know how much he bet.

It looks like you're running an elaborate statistical analysis without any solid data points to start from. Volokh is saying that Bennett and the casinos are saying different things about how much Bennett lost and he doesn't know who's telling the truth. It's hard to see how that makes Volokh a statistical illiterate. (I, on the other hand....)

Posted by: MWB on May 7, 2003 06:48 PMChip--You're exactly right--the amount one brings isn't linked perfectly with the amount one bets, though there is still a relationship.

The ultimate limits on Bennett's ability to gamble $400 million is a combination of his loss limits, the amount of time he spent gambling on each trip, and the number of trips he took.

You can do the math again and determine how much he'd have to gamble and how often to gamble $400 million.

Posted by: Thomas on May 7, 2003 07:40 PMHere's my math: assuming the "$500 slots" player can play 1 to 5 tokens per play, and consistently chooses 5, he's gambling $2500 per play.

At 1 play per minute, he takes 160,000 minutes to gamble $400M. That's 2667 hours, or about about 44 minutes a day for ten years.

At 2 ppm, it's 80,000 minutes ... 22 minutes a day for ten years.

At 3 ppm (I'm guessing this is about right for the continuous player, who hits the PLAY button again immediately on a losing play or on a small win, and maybe does a little victory dance on a win of 20 tokens or better), its 889 hours ... 89 hours a year for ten years ... maybe ten sit-down sessions, more or less ... maybe two or three casino "holidays" per year.

As some have pointed out, he might do better on video poker, especially if he's a mathematically perfect player unsusceptible to fatigue, superstition, or emotional "steaming" on losing runs. (I have no reason to believe such beings exist.)

Posted by: RonK, Seattle on May 7, 2003 08:51 PMOn a related note, suppose there is a really rare cancer (only 1% of the population has it). There is a test for this cancer that is 90% accurate. You walk into a clinic, take the test and it comes out positive. Are you more likely to have the cancer or not. Appropriate randomness assumptions apply.

Posted by: Dinesh Gaitonde on May 7, 2003 10:35 PMPlease tell me that Donald Luskin didn't actually just say that the central limit theorem is b.s. Please tell me that was actually someone pretending to be Luskin. Because otherwise someone should take away his blog.

I remember covering this stuff quite quickly and simply in a 100-level finite math class. Is it really poss. that people don't know this stuff?

Posted by: Paul on May 7, 2003 10:43 PMlies, damned lies and statistics.

but also:

it's easy to lie with statistics, it's easier to lie without it.

"Brad: That is impossible. You are 40 years old and have dropped $14,600 lifetime on lotteries (dollar a day all your life). With a standard deviation of about $1000, yours is a 15 SD event - probability is so remote that is is not even worth computing.

me: I won 40 million yesterday. Check the papers. here is my picture with the mayor."

Seeing the probability for state lotteries is around 1 in 15 million, and you only played 15,000 times, that's an extremely unlikely scenario. To put it mildly.

Posted by: Jason McCullough on May 8, 2003 12:39 AM...if he lost $8 million...impossible for him to put in $400 million ...earned $100 million for the decade.

Mark,

Assuming you make good your losses, you only need $8 million to lose $8 million. The $400 million refers to the "action" you get recycling that $8 million on the way to losing it.

Posted by: CMike on May 8, 2003 12:54 AMryan said:

I've got no problem believing that there's someone out there who HAS broken even over that length of time; I just doubt that person happens to be Bennett.

this is indeed interesting: if you believe *someone* broke even, why not Bennett? I don't like the guy either, but logically, the 'winner' (ie someone who DID break even) could be anybody who played and Bennett sure played.

And yes, you are of course correct to point out that the estimation of (2) depends almost entirely on (1). It was a badly constructed post on my part, but I think you got the gist of my argument anyway.

Bottom line, all we are doing is speculating on *whether Bennett is lying about breaking even*

In this case, Volokh is actually right: no amount of speculation will change the basic fact: either Bennett has broken even or he has not.

What if I dragged a person in front of you and said 'This guy won 3 million in the NY state Lotto yesterday'? Statistics tells you that it is extremely unlikely, but common sense tells you that *it happens every time that SOMEONE wins it* and the only question is 'is this that someone'?

with slot machines, there is no reason to believe that *someone definitely won a large payoff*, so the odds that Bennett is lying are high. But that is only partly because of the underlying low probability of breaking even. I am guessing that it is indeed possible to break even with small bets - (do slot machines have smaller payoffs with greater probability like winning 100 dollars in a lottery by matching 3 out of the 6 winning numbers?)

The real reason for scepticism is that it is very unlikely that *the same person* won enough large payoffs to total $8M - but that does NOT make it *impossible* for Bennett to be telling the truth and the *only* way to verify that is by looking at his tax records...

jason:

Seeing the probability for state lotteries is around 1 in 15 million, and you only played 15,000 times, that's an extremely unlikely scenario. To put it mildly.

This is interesting also. Check the last, oh, say, 100 lottery winners (powerball and all). I bet not one of them has dropped anything like 30,000 dollars on the lottery all their lives.

the fact that something is statistically unlikely *does not change the fact that it happens all the time*

we are definitely getting confused in the meaning of probability.

On paper, the odds of any one person winning the lottery is non-existent, practically. And yet, in every lottery *someone DOES win it*, every time.

The probability that *someone will win the lottery* is 1

The probability that it will be you is miniscule *before* the lottery

The 'probability' that it was you *after* the lottery is meaningless - the correct way to state this is:

If I guessed you won, what is the probability that I am right? That is exactly equal to the probability of you wining the lottery *before* the event (assuming I have no additional knowledge about who actually won)

Posted by: Suresh Krishnamoorthy on May 8, 2003 02:18 AMYou guys are missing something fundamental about gambling. If I go play the slots with a %90 payout, and I have $200 to gamble - How much do I go home with? $180 or $0. Almost surely the latter. There is always an upper limit restraint on how much I am willing to spend to win my money back, and the house always has the advantage.

Posted by: theCoach on May 8, 2003 06:10 AMSuresh, of all the compulsive gamblers, perhaps there's 1 (or 10 or zero) who have broken even after the numbers of times that Bennett played. Now what's the chance that this 1 (or 10 or zero) person is someone you've heard of? Someone famous?

The odds that someone has the license plate RFG 315 is pretty high. The odds that it's a famous person are negligible.

I suppose that what's really remarkable about this whole episode is that so many people with right wing leanings have spent so much time and effort (not necessarily in comments on this blog) to show that it really is possible to beat the house at gambling or at least hold your own (which is a very Freudian concept).

Posted by: jam on May 8, 2003 07:36 AMI suppose that what's really remarkable about this whole episode is that so many people with right wing leanings have spent so much time and effort (not necessarily in comments on this blog) to show that it really is possible to beat the house at gambling or at least hold your own (which is a very Freudian concept).

Posted by: jam on May 8, 2003 07:39 AMWell it looks like those of us who were highly sceptical of the $8 million net loss estimate have been proven correct as Josh Green is already revising the "estimate" downward to $1 million. Oops...

Posted by: Kevin H on May 8, 2003 08:49 AMWell it looks like those of us who were highly sceptical of the $8 million net loss estimate have been proven correct as Josh Green is already revising the "estimate" downward to $1 million. Oops...

Posted by: Kevin H on May 8, 2003 08:50 AMSo, the whole thing started with, among other things, Brad complaining about ignorance of statistics. Then, a bunch of people who demonstrated a substantial grasp of statistics set about correcting each other (including Brad)both on method and on inferences drawn. Invisible Adjunct's question about whether to accept the judgement of the nearest stats professor is pretty pointed. Now, how about trusting the judgement, just on the narrow point of how to apply statistical analysis, of those who have a background in stats, but do not teach the subject?

Posted by: K Harris on May 8, 2003 10:39 AMFull disclosure: never took statistics (although reading this thread feels a lot like it :) )

I think Suresh has nailed down exactly the problem with how almost all people deal with statistics (and why casino owners are rich). In his latest post, he laid out 3 distinct questions:

Odds of any particular person winning Lotto (simple math, based on number of tickets bought by that person vs. permutations of numbered ping pong balls; extraordinarily low)

Odds that a daily drawing has a grand prize winner (simple math, based on number of tickets bought by all people vs. permutations of numbered ping pong balls; for most games, fairly high)

Odds that someone presented as a winner actually has won - leaving aside the honesty of the presenter (simple math based on number of tickets bought by that person vs. number of tickets bought by all people and the number of winners that day; pretty low, but nothing like the odds of winning in the 1st place)

The trouble is that we perceive it as a 1 vs. 0 proposition - either this person won, or he didn't. No one would actually claim that the odds are 1 in 2, but at the back of our little instinctual primate brains, we have trouble weighing the actual numbers at hand. Maybe the odds of picking the right number were 1 in 40 million; maybe there was one winner out of 4 million tickets sold. So the odds that "this guy" won Lotto on a single ticket buy aren't 1 in 40 million anymore, but 1 in 4 million (right?). But we can't get our heads around those 7 digits, so we think, "Well, the odds are against it, but it certainly could be."

I honestly believe that our brains just can't intuitively handle this kind of statistical analysis. We can learn it, and some rare people can intuit it, but by and large, we just didn't evolve to evaluate these kinds of problems properly. Maybe our ability to hope - to envision a payoff much greater than likely - has clashed with our more animal abilities to make probability judgments. No animal would choose to stake its life (or even waste its time) on a game of 1-on-1 with Shaq - the odds are too low. But we can imagine bucking the odds, and envision the attendant glory - and we choose wrong.

My favorite statement about the foolishness of playing Powerball is that your odds of winning on a ticket that you walk one mile to buy are the same as your odds of being struck by lightning on the way - twice. But our hope makes us expend effort on one unlikely event while discounting entirely the other.

Posted by: JRoth on May 8, 2003 11:04 AMDid Bennett's end up "nearly even"?

First, the issue is semantic. What does "nearly even" mean? Obviously, "nearly even" is a polite way of saying "I lost money, but not too much". If someone turns a profit gambling, they will not say they were "nearly even". It would be difficult to argue that, for Bennett, dropping $8 mil is nearly even, but 800k over ten years is more a matter of perspective. I gamble to make money, period. But Bennett may be like most people who gamble for enjoyment and are willing to pay a certain surcharge for the thrill.

Second, we need to know how much Bennett gambled and on what types of games. We are told that Bennett gambled on slots and video poker. I would never play slots, but the odds against the higher-end players are not anywhere close to as bad as Dr. DeLong suggests and the difference between 90-94% payout and 98.5-99% payout is significant. A very studious and careful video poker player could, through selection of machines with the best odds and excellent play, actually have a positive expectation of profit. Of course, for the vast majority, fatigue, emotions, and alcohol, etc. serve to seriously degrade the quality of decision-making.

I think Volokh was correct to withhold judgement on the extent of Bill Bennett's losses. The statistical arguments are only accurate if the factual assumptions upon which they are based are accurate. I know that DeLong's assumptions with regard to slot machine payout programming are highly inaccurate. Maybe his assumption of 160,000 "pulls" is just as inaccurate. Who knows? DeLong can suppose that their is a 1/10,000 chance that Bennett lost less than $5.9 mil, but as the facts roll in, I think we will find that the number is less than a million. The bottom line is that the actual net losses suffered by Bennett depend upon the accuracy of the assumptions and facts at issue, not merely statistical analysis.

Posted by: Kevin H on May 8, 2003 11:17 AMSuresh wrote: "this is indeed interesting: if you believe *someone* broke even, why not Bennett?"

This is exactly how the NY Lotto sucks people in - with their tag line "Hey, you never know."

The problem is: you DO know. You will lose and lose and lose playing the Lotto; the exceptions are so rare - extraordinarily so - as to be nearly dismissable.

Similarly for the slots, and it only gets worse over time. The more you lose, the .... nothing. You are never "due," except for another loss. The odds are always against you.

And if you're a "high rolling" slot player (egad!) then you simply lose bigger.

After the fact: yes, SOMEONE is likely to have broken even because there are many, many people who play slots. All but one are not Bennett. Bennett is, beyond nearly all doubt, not the needle but rather one straw in the haystack. Just as is the case with any other individual - Bennett's relative fame is beside the point.

To look at it any other way is absurd. It has nothing to do with me not liking the guy. If Mother Theresa claimed to have broken even at the slots, I'd maybe give her a bit more of the benefit of the doubt, but that's just my own irrationality talking (yes, I've purchased Lotto tickets once or twice).

Only if God whispered to me that either Bennett or Mother Theresa HAD broken even would I be correct to be "agnostic" about which it was.

(Assuming, of course, that God hadn't actively chosen between them - I've no doubt who he'd reward, if it came to that.)

Posted by: ryan on May 8, 2003 11:50 AM"the fact that something is statistically unlikely *does not change the fact that it happens all the time*"

Sure, but it doesn't mean it happens to a given person all the time, which is what we're arguing here.

Posted by: Jason McCullough on May 8, 2003 12:48 PMIf you start with $10,000, and play the $500 slots, it is NOT possible for you to end up with $9800, no matter how long you play or what the odds are.

Something that people are ignoring is that he said he was "near even". If I walk into the casino with a hundred bucks, and walk out with eighty, I'm near even. If I wander in with ten grand and walk out with eight I'd be near even, too - assuming I budgeted that large a bankroll.

On a $100 denomination video poker machine - betting $500 per hand - a royal flush is worth $400,000. Half of what Bennett was losing in a year IF the $8 mil figure is right; close to half of the total lost over ten years if the $1 million figure is correct.

It isn't all that tough to play perfectly. You could memorize the table and risk a memory lapse, or you can print it out and refer to it while you play. You don't need to master the statistics behind the pay table, you just need to buy a book and pay attention to the game.

Of course Prof. DeLong is correct here.

Bill Bennett is a randomly selected gambler, chosen at a random time, and the odds that he broke even over a long time are extrememly small. He wasn't going around saying, Hey, I just won the lottery...which may or may not be true, depending on the reasons the subject may have to be saying that. It was random.

I, on the other hand, did win the lottery. But it's nothing to get too excited about.

It is unfortunate to be at or near the end of an old thread, but here, for the record, is another take on the notion that Bennett might have lost as much as reported.

Referring to "The Idiot's Guide to Gambling Like A Pro," we see that a typical video poker machine that pays back at 97.81% (a so-called 8/5-1,000 machine that pays 8 and 5 coins for a Full House and a Flush, respectively, and 1,000 coins for a Royal Flush) will have a break-even point of $2,200 for bets of five quarters per hand. That is, the average loss between Royals is $2,200 for someone playing five coins per hand in hopes of winning a progressive jackpot. A machine that is not progressive may thus only pay $250 for a Royal.

Scaling upward from $1.25 to $500 per pull, the average losses between Royals increase to $880,000. This is 1,760 hands, or about 30 hours at a minute per hand. One hopes that Bennett's machine paid a progressive jackpot for Royals, because otherwise he might only receive $100,000 for a net loss of ¾ million after four days of play. Of course, an elephant such as he might be offered a 9/6-1,000 machine with a 100.07% payback. Then the break-even becomes $488,000, and 976 hands (average) to a Royal would take only about 16 hours-three times his typical playing time per session. The classic "Inequalities For Stochastic Processes," by L.E. Dubins & L.J. Savage, considers the case when the gambler is not free to leave the casino because of the nature of the game. A few hours spent several times in a year represent a weak attempt to match and exceed the machine's average (but generally well known) performance characteristics. He is likely not obsessive enough to reach break-even over years of play. How sad: a sickness magnified by incompetence.

It is unfortunate to be at or near the end of an old thread, but here, for the record, is another take on the notion that Bennett might have lost as much as reported.

Referring to "The Idiot's Guide to Gambling Like A Pro," we see that a typical video poker machine that pays back at 97.81% (a so-called 8/5-1,000 machine that pays 8 and 5 coins for a Full House and a Flush, respectively, and 1,000 coins for a Royal Flush) will have a break-even point of $2,200 for bets of five quarters per hand. That is, the average loss between Royals is $2,200 for someone playing five coins per hand in hopes of winning a progressive jackpot. A machine that is not progressive may thus only pay $250 for a Royal.

Scaling upward from $1.25 to $500 per pull, the average losses between Royals increase to $880,000. This is 1,760 hands, or about 30 hours at a minute per hand. One hopes that Bennett's machine paid a progressive jackpot for Royals, because otherwise he might only receive $100,000 for a net loss of ¾ million after four days of play. Of course, an elephant such as he might be offered a 9/6-1,000 machine with a 100.07% payback. Then the break-even becomes $488,000, and 976 hands (average) to a Royal would take only about 16 hours-three times his typical playing time per session. The classic "Inequalities For Stochastic Processes," by L.E. Dubins & L.J. Savage, considers the case when the gambler is not free to leave the casino because of the nature of the game. A few hours spent several times in a year represent a weak attempt to match and exceed the machine's average (but generally well known) performance characteristics. He is likely not obsessive enough to reach break-even over years of play. How sad: a sickness magnified by incompetence.

It is unfortunate to be at or near the end of an old thread, but here, for the record, is another take on the notion that Bennett might have lost as much as reported.

Referring to "The Idiot's Guide to Gambling Like A Pro," we see that a typical video poker machine that pays back at 97.81% (a so-called 8/5-1,000 machine that pays 8 and 5 coins for a Full House and a Flush, respectively, and 1,000 coins for a Royal Flush) will have a break-even point of $2,200 for bets of five quarters per hand. That is, the average loss between Royals is $2,200 for someone playing five coins per hand in hopes of winning a progressive jackpot. A machine that is not progressive may thus only pay $250 for a Royal.

Scaling upward from $1.25 to $500 per pull, the average losses between Royals increase to $880,000. This is 1,760 hands, or about 30 hours at a minute per hand. One hopes that Bennett's machine paid a progressive jackpot for Royals, because otherwise he might only receive $100,000 for a net loss of ¾ million after four days of play. Of course, an elephant such as he might be offered a 9/6-1,000 machine with a 100.07% payback. Then the break-even becomes $488,000, and 976 hands (average) to a Royal would take only about 16 hours-three times his typical playing time per session. The classic "Inequalities For Stochastic Processes," by L.E. Dubins & L.J. Savage, considers the case when the gambler is not free to leave the casino because of the nature of the game. A few hours spent several times in a year represent a weak attempt to match and exceed the machine's average (but generally well known) performance characteristics. He is likely not obsessive enough to reach break-even over years of play. How sad: a sickness magnified by incompetence.

Without friends no one would choose to live, though he had all other goods.

Describing is not knowing.

Posted by: Lewin Luigi on December 20, 2003 01:00 PMI have become Death, the destroyer of worlds.

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