Problem 30 - A First Course in Probability

Question

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?

Step-by-Step Solution

Answer

The expected value of the player's winnings in the St. Petersburg paradox is \(\infty\). However, despite the infinite expected value, many people would not be willing to pay $1 million to play the game once or repeatedly, as the most probable outcomes provide much lower winnings compared to the cost, and the small probability of extremely large winnings does not compensate that risk. If allowed to play for as long as desired, some players may be more willing to pay the cost, although the potential for significant loss remains a deterrent.

Step 1: Compute the probability of each outcome

Let's first find out the probability of the tail appearing on the \(n\)th flip. Since the coin is fair, the probability of getting tails on any single flip is \(0.5\), and the probability of getting \(n-1\) heads in a row before the first tail appears is \((0.5)^{n-1}\). For example, if \(n=3\), the probability of the outcome is \(0.5^2 = 0.25\) because there are two coin flips with heads.

Step 2: Calculate the winnings corresponding to each outcome

For each outcome, we are given that the person wins \(2^n\) dollars. Given a certain number of flips, \(n\), we can calculate the winnings as \(2^n\) dollars.

Step 3: Compute the expected value of the winnings

The expected value of the player's winnings is the sum of the expected winnings for each outcome. For each outcome, we multiply the probability of the outcome with the corresponding winnings. Mathematically, we can write this as: \[E[X] = \sum_{n=1}^{\infty} P(\text{tail on }n\text{th flip}) \times (\text{winnings on }n\text{th flip})\] \[E[X] = \sum_{n=1}^{\infty} (0.5)^{n-1} \times (2^n)\] Next, let's simplify the expression above: \[E[X] = \sum_{n=1}^{\infty} (\frac{1}{2})^{n-1} \times 2^n\] \[E[X] = \sum_{n=1}^{\infty} 2\] Since the summation diverges, the expected value of the player's winnings is \(\infty\). This means that, theoretically, over an infinite amount of games, the player should have infinite winnings, which is referred to as the St. Petersburg paradox.

Step 4: Answer questions (a) and (b)

(a) Considering that the expected value of the winnings is infinite, it might seem that paying \(1\) million dollars to play the game once is a good idea. However, people usually refuse this offer because the potential loss of \(1\) million dollars is significant. Moreover, the most probable outcomes of the game provide much lower winnings compared to the cost, and the small probability of extremely large winnings does not compensate that risk. (b) If a player had the option to play the game for as long as they liked and only settle up when they stopped playing, they might be more willing to pay \(1\) million for each game. In this scenario, the chances of having a large winning increase as the number of games also increases, and it might help players to offset the costs. However, people still may not be willing to take the risk, as the potential loss could be significant if they need to stop playing earlier than they planned.