Paradoxical expected values

We tend to think of the expected value of a random variable as a standard, practical, and simple measure of the variable’s central tendency.  However, there are situations in which the expected value is misleading or even nonexistent.  One famous case, commonly called the “St. Petersburg Paradox”, illustrates perfectly the limitations of the expected value as a measure of central tendency.

Suppose a fair coin is flipped until the first tail appears.  You win $2 if a tail appears on the first toss, $4 if a tail appears on the second toss, $8 if a tail appears on the third toss, etc.  The sample space for this experiment is infinite: S = {T, HT, HHT, HHHT, …}.  A game is called fair if the ante, or amount you must pay to play the game, is exactly equal to the expected value E(x) of the game.  Casino games are obviously never fair, because casinos stand to earn a profit (on average) from the games they offer – otherwise, they would not have a viable business model and they couldn’t afford to give you free drinks and erect fancy statues (though, while we’re on the subject, craps is the “fairest” as far as casino games go in the sense that the odds are skewed least heavily in the dealer’s favor).

Let the random variable W denote the winnings from playing in the game described above.   We want to categorize W such that we can mathematically evaluate its expected value.  Once we find this expected value, we know how much we’d have to ante for this game to be considered fair.  We know how much we’ll win on the first, second, and third toss, but let’s generalize for k tosses:

  • T on 1st toss -> $2
  • T on 2nd toss -> $4
  • T on 3rd toss -> $8
  • T on kth toss -> $2^k

So we win $2^k if the first tail appears on the kth toss.  But what is the probability of each of those outcomes?  For toss one, there is a ½ chance of getting a tail.  Since trials are independent, the probability of the first tail appearing on the second toss is (1/2)*(1/2) = 1/4.  For the third toss, the probability is 1/8.  So the probability of winning $2^k is the probability of getting the first tail on the kth toss which is:

Screen shot 2013-11-01 at 7.01.14 PM

To find the expected value of the random variable W, we need to find the sum over all values that k (remember that k = the number of tosses before the first tail) of k times the probability of k.  It’s evident from the sample space that k can take on an infinite number of values.  The expected value is:

Screen shot 2013-11-01 at 7.01.21 PM

So the expected value of the random variable W (winnings) is the sum over all k of the payoff (2^k dollars) times the probability of realizing that payoff (p(2^k)).  So the first term is the actual payoff, the second is the probability of that payoff.  Plugging in what we know about the probability of k and the infinite nature of the sample space:

Screen shot 2013-11-01 at 7.01.30 PM

The sum diverges!  There is no finite expected value.  That is, the expected value of the game is infinite, which means we’d have to pay an infinite amount of money for the game to be fair.

The point is not that you should wager an infinite amount of money in order to play the aforementioned game.  The point is that the expected value is often an inappropriate measure of central tendency that leads to an inaccurate characterization of a distribution.

About schapshow

Math & Statistics graduate who likes gymnastics, 90s alternative music, and statistical modeling. View all posts by schapshow

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