Out of nothing, a man appears in front of you. With a coin in one hand and an infinity bag (a bag with a finite area and an infinite volume) filled with banknotes in another, he asks you the following.
"Dear Madam, I will flip a coin - and it is a fair coin, this, I promise -, and if it flip heads, you will receive two rubles and another flip from me. If it is tails, you will walk away with the two rubles you gained in the first round. If it is heads again, your gain doubles and you get four rubles more, and another flip, in which, if the coin flips heads again, you receive the double of four, that is, eight rubles. And so on, to infinity - and there is an infinite amount of money in my bag, this, I promise. How much would you be willing to pay to enter in the following game with me?"
It is easy to see, that the expected value of this probability game is equal to infinity. EV = 1/2 * 2$ + 1/4 * 4$ + 1/8 * 8$ + ... = ∞ $. The reason is that while each succeeding head-flip is less probable than the one before (at the time of the bet, not at the time of the throw of the coin, because before each throw, the probability of heads is, of course 50%!), but the increasing gains of each succeeding flip make the expected value of every flip 1 $ (1/2 * 2$ = 1$, 1/4 * 4$ = 1$, ...). This makes this series a sum of one dollars to infinity - 1$ + 1$ + 1$ + 1$ + ... = ∞ $.
If you would be rational, you would then be willing to pay the man an amount just short of infinity (whatever that is). In reality, people are willing to give around five, six, at most up to ten dollars. This is what economists call St. Petersburg paradox, but maybe it is better to call it a prescriptive and a descriptive failure of the assumption of rationality.
Author: Tej Gonza