Associated Topics || Dr. Math Home || Search Dr. Math

### Fair Value of a Coin Toss Game

```
Date: 01/21/2001 at 17:19:00
From: Raj Shah
Subject: Pricing a coin toss game

Hello. Here's my question:

Suppose you play a game where you toss a coin untill Tails turn up.
The payout of the game is 2^n, where n represents the number of
'Heads' that come up before a 'Tails'.

For example, if you flip the sequence: H H H T, then the payout would
be 2^3, or 8 dollars. If the first toss is a Tail, the payout is 2^0
or 1 dollar.  I'm trying to find the fair value of the game.

I started doing an expected value calculation, figuring the
probability of each sequence multiplied by the payout.

T :     Probablity = 1/2 * 2^0 = 1/2
H,T :   P = (1/2*1/2) * 2^1 = 1/2
H,H,T:  P = (1/2*1/2*1/2) * 2^2 = 1/2

Now I'm stuck. I get an infinite sequence of 1/2+1/2+1/2,.... What
does this imply?  I'm trying to figure out the "fair value" of this
game, or what you would pay to play the game over time. I would really
appreciate your help, as I've been stuck for some time.

Thank you,
Raj
```

```
Date: 01/22/2001 at 15:13:14
From: Doctor Twe
Subject: Re: Pricing a coin toss game

Hi Raj - thanks for writing to Dr. Math.

This problem is called the St. Petersburg paradox. To determine a
"fair value" (also called the expectation) for the game, we multiply
the probability of each outcome by the payout, and sum the results. As
you noted, the probability of getting the first tail on the kth toss
is:

P(k) = (1/2)^k

and, with a payout of 2^(k-1), we get an expectation of:

inf.                      inf.
E(X) = SUM [(1/2)^k * 2^(k-1)] = SUM [1/2] = 1/2 + 1/2 + 1/2 +
...
k=1                       k=1

which adds to infinity. This means that over the long haul, no matter
what price you set to play, the player will ultimately win money - in
theory. In practice, even for small values of k, winning is highly
unlikely, and having the patience (and money) to play the required
number of games to win is unlikely. For example, to win \$2^30 (just
over \$1 billion), you should get 30 heads in a row followed by a tail.
The chances of this happening are less than 1 in 2 billion (1 in
2,147,483,648 to be exact.)

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search