Unexpectedly Great Expectations Post

here shows an awesome paradox. The paradox is called St. Petersburg paradox. I have spent hours explaining this paradox to many smart friends of mine. This is probably the best paradox I have ever seen.

Just reposting the game from that article:

**WARNING: THESE ARE THEORETICAL GAMES. Try not to bias yourself by how much YOU value 1000$ compared to how a millionaire values 1000$ (the utility function of money is a constant for all people). Also, we assume God has infinite amount of money with him, and does not lie when he says he will pay you, so please don’t give arguments like “put the money on the table and I will play” (replace 1$ by 0.001$ or any such figure if you want to satisfy yourself practically).**

God offer you the option of playing a game, *exactly once*, against me. This is how the game works. God will toss a fair coin until a T turns up. The sequence of coins H^{n}T will earn you 2^{n} dollars. More explicitly, a T on the first toss gives you 1 dollar, a Head followed by a Tail gives you 2, HHT gives you 4, HHHT gives you 8. As soon as the T turns up, we settle accounts, and leave, never to see each other again. However, there is a constant pre-agreed charge P you must pay to play this game against me (say 10000 $). Upto what price P are you willing to play this game?

*Analysis:* The probability of the T is 1/2, of HT is 1/4, of HHT is 1/8 and so on (1/2 * 1/2 * 1/2… as they are independent events).

Hence your expected value of earnings for this game,

E(earnings)

= P(T).Earnings(T) + P(HT).Earnings(HT) + P(HHT).Earnings(HHT)….

= (1/2 * 1) + (1/4 * 2) + (1/8*4) + (1/16*8) + …. = 1/2 + 1/2 + 1/2 + 1/2 …. = (infinite).

However the 2000th term of this series of halves is highly improbable (1/2^1000). If you believe expected values, you should be willing to pay any finite amount of money to play this game.

But if you think over it, the probability that you will get at least 1000$ is 0.0005 which is too small. So, you should not be willing to pay infinite amount of money. Your intuition will not allow you to play with infinite money. Can you explain the paradox?

**Solution:** The wikipedia article on

St. Petersburg paradox