A person tosses a fair coin until a tail appears for the...

A person tosses a fair coin until a tail appears for the first time. If the tail appears on the $n$ th flip, the person wins $2^{n}$ dollars. Let $X$ denote the player's winnings. Show that $E[X]=+\infty .$ This problem is known as the St. Petersburg paradox. (a) Would you be willing to pay $\$ 1$ million to play this game once? (b) Would you be willing to pay $\$ 1$ million for each game if you could play for as long as you liked and only had to settle up when you stopped playing?

A person tosses a fair coin until a tail appears for the first time. If the tail appears on the $n$ th flip, the person wins $2^{n}$ dollars. Let $X$ denote the player's winnings. Show that $E[X]=+\infty .$ This problem is known as the St. Petersburg paradox.
(a) Would you be willing to pay $\$ 1$ million to play this game once?
(b) Would you be willing to pay $\$ 1$ million for each game if you could play for as long as you liked and only had to settle up when you stopped playing?

Key Concepts

Expected Value

The expected value is the weighted average of all possible outcomes in a probability scenario, where each outcome is multiplied by its likelihood. In the context of the paradox, each possible win is weighted by its probability, leading to an infinite sum even though typical wins are small. This concept is central in decision theory and risk analysis when evaluating potential gains or losses.

St. Petersburg Paradox

The St. Petersburg paradox is a classical problem in probability theory that highlights a discrepancy between mathematical expectation and human intuition. It involves a game with a payout structure that produces an infinite expected value, yet most participants are unwilling to bet large amounts due to the low probability of receiving a high payout. This paradox has wide implications for decision-making under uncertainty.

Divergence of a Series

In this problem, the calculation of the expected winnings involves summing a series where the contribution of each term does not diminish; instead, each term effectively contributes a positive constant to the sum. The divergence of this series, where each term equals one, mathematically explains why the expected winnings are infinite, despite the overwhelming likelihood of low outcomes.

Utility Theory and Risk Aversion

Utility theory addresses the disconnect between the mathematical expected value and individual preferences in decision-making, incorporating the principle of diminishing returns where each additional unit of wealth delivers less added satisfaction. Risk aversion explains why, despite an infinite expected value, people would be reluctant to pay large sums to engage in the game. This concept is crucial in economics and finance, influencing how real-world decisions are made under uncertainty.

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