Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Calculus or Analysis
    Number Theory
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Advanced level:

St. Petersburg Paradox

A new casino offers the following game: you toss a coin until it comes up heads. If the first heads shows on the N-th toss, you win 2N dollars. (Thus the payoff doubles with each coin toss that isn't heads.)

How much should you be willing to pay to enter this game?

At first glance, you might think it is the expected value of the payoff to the player. But if you calculate it, you get a divergent series... the expected value is infinite!

If so, then maybe you should be willing to pay any fixed finite amount of money to play this game? And yet the chance of winning more than 4 dollars is only 1/4, so that can't be right, can it?

Presentation Suggestions:
Alternately, you might ask students, if they were the casino owner, how much they should be charge to play this game? Or, would they even offer the game?

The Math Behind the Fact:
The more general question is: how should players evaluate their preferences over options that involve chance? (Classic examples are games, but such options could occur in any decision that you face in your life.) Naively, you might think that you should choose the option that gives you the highest expected value.

The lesson of this paradox is that people (like yourself) do not play games as if they are maximizing the expected monetary value they receive. However, certain rationality assumptions about the way people behave (the von Neumann and Morganstern axioms) do imply that people do act as though they are maximizing something, which is often called a utility function. See Deal or no Deal for an idea of how to construct a utility function. This subject is often treated in detail in a course on game theory, which is the mathematical modeling of decision-making.

How to Cite this Page:
Su, Francis E., et al. "St. Petersburg Paradox." Math Fun Facts. <>.

Keywords:    game theory, probability
Subjects:    probability
Level:    Advanced
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!