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A person tosses a fair coin until a tail appears for the first time. If the tail appears on the nth flip, the person wins 2n dollars. Before the nth flip, he does not receive nor pay anything. Let X denote the player's winnings. Show that E(X)=&#61537;&#61486;&#61472;Would you be willing to pay \$1 million to play this game? Would your answer change if you were and idle billionaire?
This is called the St. Petersburg Paradox.

#### Solution Preview

Probability that the first tail occurs in nth toss is:
P(n) = P(Head on first toss) * P(Head on second toss) * ......P(Tails on nth toss)
As the coin is a fair coin, the probability of Head or Tail = ½
Thus, P(n) = ½ * ½ * ............* ½
P(n) = ½n
The Player earns \$2 if the tail occurs in 1st toss. As the probability of tail is ½ in any toss, the expected return if the tail occurs ...

#### Solution Summary

The St. Petersburg Paradox is analyzed. The expert determines if they would be willing to pay \$1 million to play this game.

\$2.19