Quote Originally Posted by warty goblin View Post
Ah, probability confusion! Something I'm actually somewhat qualified to talk about!

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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.
It does, thank you.

The "gambler's fallacy" is the assumption that the next event is dependent when, in fact, it is independent. So a bet on a coin flip should be made without regard to previous history, assuming a fair coin.

By contrast, the casino isn't using the gambler's fallacy to predict a single event . It is, instead, playing the averages. It can reasonably expect that, if the probability of a casino win is 0.75 and a player win is 0.25, then the casino will make money *in the long run*, because the averages will smooth out deviations caused by someone being 'lucky'.

The casino is correctly using probability to predict an expected outcome over a statistically significant sample. The gambler is not, because he is attempting to use probability that works over long averages to predict a single event.

Incidentally, I once did some research to find out how professional gamblers make their living. Short answer: By having a great tolerance for debt and by being subsidized by casinos, as a professional gambler can be a sort of marketing stunt, since he advertises by example that gambling can be a 'success', even though it isn't.

Respectfully,

Brian P.