Quote Originally Posted by pendell View Post
I'm not sure I follow. You're right in that this is what gambling houses do. I'm told that casinos calculate their expected earnings for a given table thus:

Revenue = House Edge * Bets.

Where "bets" is the total money put down and "house edge" is the probability that the house will win any given bet.

So from a Casino's point of view, the gambler's fallacy would encourage me. If I were to walk the floor and see someone winning twenty hands of blackjack in a row, I would first send Vinnie and Louis the Louse to ensure he wasn't cheating. Once I was sure this really was the result of blind luck, I'd let him continue playing. After all, the laws of probability can't be mocked forever. If he keeps playing at that probability, not only will I eventually win it all back but I'll get his starting stake too. It may not happen on the next hand or the next twenty hands. It may take several years, but it will happen. Meanwhile the winnings will encourage OTHER people to put down money at bad bets, resulting in even more cash for me.

How am I , as the Casino owner, not succumbing to the gambler's fallacy?

This leads up to my sure-fire foolproof system to winning at gambling: Buy a casino and make the laws of probability work for, rather than against, you :).

Respectfully,

Brian P.
Ah, probability confusion! Something I'm actually somewhat qualified to talk about!

So your confusion from your first post stems from not taking conditional probabilities into account.

In your first example, you are correct that the odds of your next flip being heads is 1/2. You are also right that your probability of getting 49 heads in a row is very, very bad. But your chances of getting 49 heads in a row given you have gotten 48 heads in a row is the same as the probability of getting heads on any flip of the coin: 1/2

You can think about this intuitively. At this point in time, you have flipped 48 heads. After the forty-ninth flip, you can have either of two outcomes: 49 heads in a row, or 48 heads followed by one tail. Since heads or tails are equally likely, both have probability 1/2 of happening, given you already have the 48 heads. Ergo, given 48 heads, the probability of getting a 49th is 1/2.

In more theoretical terms, conditional probability for events is defined as P(A | B) = P(A and B)/P(B), where A is any event (such as getting 49 heads), B is any event with non-zero probability. The | is read 'given' here.

The intuition for this weird definition is that since B has happened, you are only interested in the parts of A that can happen concurrently with B. You divide by B because you have to adjust for the change in P(A and B) due to B having happened - intuitively you are counting the number of outcomes in A and B, then dividing by the number in B. (This last sentence is technically incorrect in many cases, but don't worry about that here)

So for this case, let A be 49 heads, B be 48 heads. Then P(A|B) = P(49 heads and 48 heads)/P(48 heads) = P(49 heads)/P(48 heads) = (.5)^49/(.5)^48 = 1/2.

Take-away: You aren't betting on getting 49 heads, you're betting on getting one more head.

OK, for your second confusion, casinos care mostly about expected values, which are probability weighted averages. Simply put, you take each outcome of a game, multiply that by the probability, and add 'em all up. If all outcomes are equally likely, this is just your bog standard arithmetic mean. It's easiest to think about it as the value the average winnings approach as you play lots and lots of games.

So here's an example. Suppose the buy-in for a game is $5. Suppose furthermore than the gambler will win $10 with probability 1/4, and therefore lose with probability 3/4. The casino's expected income is
5*P(gambler loses) - 10*P(gambler wins)
= 5(3/4) - 10(1/4) = 5/4, or $1.25

Assuming the casino has a reasonable amount of money in the vault, it won't care about whether or not somebody wins or loses an individual game. It won't even matter if somebody always wins (as long as they aren't cheating). They are, on average, making money.

The difference between this and the first example is that the casino is basically playing the long game. The gambler only cares about the next flip. You shouldn't gamble not because you will lose money, but because you will probably eventually lose money, and the more you gamble, the more likely it is that you come out behind.

Hope that helps.