Q.4.30 - 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 2n dollars. Let X denote the player's winnings. Show that $E\left[X\right]=+\infty$. 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

(a) The individual is not ready to pay $ 1 million to play this game once. 

(b) The individuals will never be willing to pay a $ 1 million to play this game. 

Step 1: Given information (Part a)

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 2n dollars 

Step 2: Solution (Part a)

From the given data, a person throws a fair coin until a tail occurs for the first time. If a tail appears on the first toss, the person wins 2 dollars.

If a tail appears on the second toss, the person wins $2^2$ dollars.

If a tail appears on $n^{\text{th }}$ toss, the person wins $2^n$ dollars.

Then,

From the comprehended information, the probability that obtaining tail is $\frac{1}{2}$.

The probability that tail on first toss $\left(T\right)=\frac{1}{2}$.

The probability that tail on second toss $(H,T)=\frac{1}{2}\times \frac{1}{2}={\left(\frac{1}{2}\right)}^2$.

The probability that tail on n th toss, $(H,H,\dots ,T)=\frac{1}{2}\times \frac{1}{2}\times \dots \times \frac{1}{2}={\left(\frac{1}{2}\right)}^n$.

Let X be the event that player's winnings.

Here, the range of X is, X=1,2,....

So the expected winning of the player will be:

$E\left(X\right)=\left(2\times \frac{1}{2}\right)+\left(2^2\times \frac{1}{2^2}\right)+\left(2^3\times \frac{1}{2^3}\right)+\dots .$

$=\sum _{n=1}^{\mathrm{\infty }} \left\{2^n\left(\frac{1}{2^n}\right)\right\}$

$=\sum _{n=1}^{\mathrm{\infty }} 1$

$=\mathrm{\infty }$

Step 3: Solution (Part a)

The above-mentioned information tells that, the probability of winning 2 dollars is 0.5,4 dollars is 0.25,8 dollars is 0.125, and so on.

So, the probability of winning $ 1 million is very minor. Existing a risk element, an individual would pay $ 1 million because there is a possibility for loss.

Step 4: Final answer (Part a)

Thus, the individual is not ready to pay $ 1 million to play this game once.

Step 5: Given information (Part b)

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 2n dollars 

Step 6: Solution (Part b)

If an individual is willing to pay $ 1 million for per game, then the probability of winning a million is ${\left(\frac{1}{2}\right)}^n$.

So, the amount that the person will win for each game will be,

$2^n\ge 1000000$

(Apply log on both sides)

${\log }_{10}^{2^n}\ge {\log }_{10}^{\left(1000000\right)}$

$n{\log }_{10}^2\ge 6$

$n\ge \frac{6}{{\log }_{10}^2}$

$n\ge 19.93$

$n\approx 20$

The probability of winning a million is,

${\left(\frac{1}{2}\right)}^n={\left(\frac{1}{2}\right)}^{20}$

$=0.0000009537$

$\approx 0$

Step 7: Final answer (Part b)

So, the probability of succeeding a million is almost equal to zero. Thus, the possibility that people willing to pay $ 1 million to play this game is almost 0. Thus, the individuals will never be willing to pay a $ 1 million to play this game.