Just reposting the game from that article:
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,
= 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