My second course for The Great Courses has been recently released (shameless plug) -Your Deceptive Mind. It's a skeptical romp through the workings of the human brain, and a good primer of skeptical thinking.
In the course I cover the topic of innumeracy - the fact that humans have a poor innate grasp of math and statistics. I expected to get a disproportionate amount of feedback from these two lectures, based on my experience with the SGU. The topic that has lead to the greatest number of e-mails over the years has been the Monty Hall problem.
OK, quickly, here it is again for those who don't know. You are on the show, Let's Make a Deal, and you are offered three doors. Behind two doors there is a goat, behind the third is a new car. Monty Hall, the host, knows what is behind each door. After you pick a door Monty Hall opens one of the other doors to reveal a goat (which he always does), and then asks if you would like to switch your pick to the other remaining door. What should you do? The answer is that you should switch, because it will increase your chances of winning from 1/3 to 2/3. Some people have a hard time grasping the statistics involved - and that's the point.
I discuss the Monty Hall problem in the new course and warned the publisher that this will likely result in a great deal of feedback. However, my first e-mail regarding innumeracy concerns a separate issue. In the course I claim that if the odds in a casino were exactly even then the casino would still make money over the long haul. The reason is this: if the odds are truly even and fair, and we assume that magic, lucky charms, prayer, and psychic powers do not work (just for the sake of argument), then individual winning and losing will follow a drunkards walk of randomness. This will average out over any stretch of time, with the error bars narrowing to zero over longer stretches of gambling.
However, there is an asymmetry to this process - the point at which people stop gambling. In order for winning and losing to average out across all gamblers for the casino, there must be no asymmetry in when winners and losers stop gambling. If gamblers stop at a random point, or after a set point (by time, number of bets, or number of free drinks - whatever), something that has nothing to do with whether or not the gambler is currently winning or losing, then the casino will break even over time.
However, the above scenario is not the case. There is an asymmetry. Some gamblers go bust. They lose all their money (or all the money their spouse lets them bet) and then are forced to stop gambling. The house, however, never goes bust because their pockets are just too deep, and there are limits on how much individuals can bet. You can describe this as most gamblers having an absorption wall on the losing side of their drunkards walk of winning and losing.
I did not think that this claim would turn out to be controversial, but I recently received the following e-mail:
"In Lecture 11 of the Critical Thinking Series, around 5 minutes in, you discuss the Absorption Wall, and state that the casinos would still rake in lots of money because of it."
"An Absorption Wall is there alright, but it has absolutely no effect on the casino winnings. If the Expected Value of a fair coin toss is $0,00, it will still be 0 with a player that has been losing or a player that has been winning, as you've said yourself many times before explaining other fallacies. But then you go on to confound regression to the mean thinking that if a winning player keeps playing, he will go back to his average of zero dollars won in the long run. But the regression to the mean will make sure that his average winning per bet, in all his sample size, will keep decreasing and reach 0 at infinity, but not his absolute winnings. In fact, if he won $5000 so far, he has exactly 50% change of having more than $5000 after 100 more bets; what will probably go down is his average winning per bet, not his absolute winnings."
"I made a program to simulate that just to make sure I was not crazy or missing something, by the way, which confirmed I was not. After several minutes simulating, the casino busted more than 9k gamblers, but is still losing 200k for an average of -$0.00006 per 10 dollar bet... Some lucky gamblers have a lot of money, and while their average winnings per bet are appropriately approaching 0 as they keep playing, the money they won so far does not."
First, I do agree with many of his premises. It is true that once someone has won $5000, if you start tracking statistics at that point then $5000 becomes his new baseline and he is 50% likely to have more or less than $5000 at some arbitrary point in his gambling future. This reflects the fact that each bet is an independent event, and does not "remember" the result of previous bets. I also agree that regression to the mean affects average winnings, and not total winnings - if we reset our baseline.
I admit there is more than one way to look at this problem, with different approaches giving different intuitive answers. The e-mailer is looking at the problem this way – the absorption wall removes gamblers from the pool of all gamblers (because they go broke). However, from the point at which they are removed they are just as likely to win and regain their losses as they are to lose even more money, therefore their removal has no effect on the Casino’s net profits.
You can look at the problem another way, however. Because of the absorption wall each individual gambler is more likely to walk away a loser than a winner. Think about it this way – if you had $10 to gamble in Vegas and were vacationing there for a week, what are the odds that you would go home a winner? It seems obvious that it’s pretty small, as the chances that you would go bust are pretty high with such a small starting amount.
The same is true, however, of larger amounts of money – hundreds or even thousands of dollars, even though the chance of going bust is smaller than if you start with $10.
The absorption wall of going bust, therefore, makes it more likely that you will go home a loser than a winner, and since you are playing against the house it makes sense that the house has a greater chance of being a winner than a loser.
This effect could be offset if winners win more than losers lose, to compensate for the greater number of losers. This may be true. The absorption wall has two effects - it increases the chance of going home a loser, but it also sets a limit on how much an individual can lose. There is no practical limit on how much a winner can win, however, since the casino has too much money to go bust.
This latter view is based on a classic statistical problem known as the gambler’s ruin. In this problem you have two gamblers each with different amounts of starting money. They will compete against each other in a game of chance until one goes broke. The problem is to devise an equation that will give the probability of either gambler winning.
The statistics here clearly show that the gambler with the larger amount of starting money is more likely to win. In statistical discussions of the gambler’s ruin the implication for casinos is often raised.
And that is why you are statistically doomed to lose in Las Vegas. Even if the odds are not against you, unless you can match the resources of the house you will eventually end up on the short end.”
But this is not an exact analogy to the question at hand – because in this statistical model the gambling continues until one person loses everything. If that were the case, then you would have no chance against the house. But that is not how Vegas works for most people. Most gamblers are visiting for a finite period of time, so they will stop gambling when they either run out of money or go home (again, assuming they do not self-impose a stopping point).
The e-mailer also mentions a computer simulation, which could be a nice way to solve the dilemma. However, the details of the simulation are important. Are gamblers that go broke replaced from an infinite pool of new gamblers, or is there a finite pool of gamblers who have multiple stopping points - going home or going bust. Then again, perhaps it doesn't matter, but that is the very thing we would be testing.
Here is a related thought experiment. The gambler's ruin equation states that, with a finite bankroll, every gambler will eventually go broke. So let's say we have a hypothetical casino in which gamblers enter with a finite stake and always play until they go broke. When they do they are eliminated and replaced by another gambler with a finite stake. Therefore all gamblers enter the casino with a finite amount of money and leave with no money.
It seem that in such a scenario, even with 50/50 odds, the casino must be making money. But I suppose it's possible the influx of money is not going to the casino but is being held by temporary winners.
Here is one more thought - let's say you flip a fair coin a random number of times. With more flips the outcome approaches the predicted outcome of 50% heads and 50% tails. But let us also say that you employ one rule - you must always stop after flipping heads. Wouldn't that bias the statistics in favor of heads, even a little bit?
Likewise, if a gambler goes broke, then it stands to reason that they lost their final bet. Wouldn't that also bias the win-loss ratio toward the gambler losing? Would this explain how the casino could win even with 50/50 gambling odds?
Conclusion
After researching this problem further and thinking it through as above I now believe that the e-mailer may be correct. I could not find a definitive answer online - but I did find many sites repeating my original claim, that casinos would still make money even with even odds.
Sites seemed to be split between those citing my original logic, and those citing the e-mailer's logic. That logic, I admit, is compelling - every bet has even odds and so is statistically neutral for the casino, regardless of who is making the bet, their current bankroll, and their prior history of winning and losing. Each bet is a statistically independent event.
But I am not convinced that the phenomenon of going bust does not create a statistical bias toward gamblers losing, perhaps for the reason I stated above (always losing the final bet).
So I leave it my readers to further explore this dilemma and see if we can come up with a definitive solution.
I guess I can take some solace in that the whole controversy proves my original point - people generally lack an intuitive sense of probability and statistics.
Steven Novella, M.D. is the JREF's Senior Fellow and Director of the JREF’s Science-Based Medicine project.
Dr. Novella is an academic clinical neurologist at Yale University School of Medicine. He is the president and co-founder of the New England Skeptical Society and the host and producer of the popular weekly science show, The Skeptics’ Guide to the Universe. He also authors the NeuroLogica Blog. |