The St. Petersburg Paradox at CTY

This week, I am at my Johns Hopkins Center for Talented Youth Program, and I am taking Game Theory and Economics. Like last year, I wanted my post to be something cool I had learned that week.

First off, there is something called expected value, which means that if you do the experiment say millions of times and average all of your results, you will be extremely close to the expected value.  To determine expected value, you must multiply each outcome by the probability of that outcome occurring. Add all of those up and you have the expected value.

EV = P(a) x a + P(b) x b + ...

For a standard die roll, the expected value is:

EV = 1/6(1) + 1/6(2) + 1/6(3) + 1/6(4) + 1/6(5) + 1/6(6) = 3.5

So, the expected value is 3.5. Of course, you cannot get 3.5 on a single die roll, but if you average together a thousand rolls, you are sure to be near 3.5.

As an expected value problem to solve, our teacher gave us what is called the St Petersburg Paradox, which describes a mathematician who is told that he can play a game where he flips a coin until he gets tails. Afterwards, he will get 2^n pieces of gold, with n being the number of heads he flipped before he flipped a tails. The mathematician immediately went to determine the expected value, which is:

EV = 1/2(1) = 1/4(2) + 1/8(4) + ... EV = 1/2 + 1/2 + 1/2 + ... = infinity

This is saying that if you average all of your trials, you will get infinity. However, you have a fifty-fifty chance of getting just one piece. How can this be?

As I mentioned earlier, this is a paradox; it's mathematical answer differs from its logical answer. For a logical answer, you can cut it off at how much gold they are able to award him (they can't afford 2^50 pieces) and calculate. It still won't be very accurate until huge numbers of trials. Anyways, I found that pretty cool.

Bonus: We recently received a puzzle as well. As usual, I will provide the answer in a month.

By pure random guessing, what is the probability that you will get this answer correct.
  a. 50%
  b. 25%
  c. 0%
  d. 50%