Greg Charles wrote:

“My probability professor taught us that Las Vegas casinos would make money even if the odds weren’t stacked in their favor. Why? Because people dream of “breaking the bank”, and generally this would meaning winning more than they have to lose. For example, you go to Vegas with $500 and promise to stop playing when you run out. (Disregard addiction for the moment.) Generally, you don’t put an upper limit on winnings. Even if the odds were exactly even, you would be very likely to hit your lower limit at some point and have to stop, while you may have hit $500 in winnings earlier, but decided to keep playing.”

Greg, I hope your memory is faulty, because this concept is fallacious. One way to see this is to look at things from the casino’s point of view. Suppose the house has no edge. Then, on any round of betting, the casino wins as much as they lose. The dice and cards and roulette wheels don’t know which customer is over or under some limit. So, the casino breaks even and the customers break even (on average).

Try enother explanation: Suppose that you are betting on flips of a coin. You win or lose $1 depending on how the coin comes up. At any point in time, your expected (average) result is zero. It just doesn’t matter what your stopping rule is.

A similar conundrum involves the sex of new babies. Suppose that parents in some country are partial to boys, so they continue having babies until they have at least one son. It might seem that more boys than girls would be born. However, bear in mind that each birth is a 50-50 of either sex, so in fact the same number of each sex is born. (I am ignoring the fact that babies are slightly more likely to be boys, but am assuming 50-50.)