# Expected value

(In memory of Toxie 😀 )

Let’s suppose the following bet were proposed:

• The bettor pays $100 to the house. • An honest coin is tossed until a head appears. • The bettor receives$ 2n from the house, being n the number of coin tosses.

Is this bet worth making? Why?

An analysis will be given in the next post.

## 3 thoughts on “Expected value”

1. Demian says:

Interesting.

My first not-very-thoroughly-thought guess would have been: “of course that bet is not worth making, as the expected value of N, the number of coin tosses, is 2”.

But, correct me if I’m wrong, it would seem that the random variable M = 2 ^ N, the money gained each bet, does not have an expected value. As (I hope to the LaTeX embbeding thing right)
$\sum_{n=1}^{\infty} p(n) * 2^n = \sum_{n=1}^{\infty} 0.5^n * 2^n = \sum_{n=1}^{\infty} 1$
diverges.

Then, Ive never thought about the mening of a random variable not having an expected value before.

I run some tests and it seems that the bet is not worth it (good… my intuition was not that wrong =P). However, half-baked tests are not a very good proof. The results, besides, are quite random, even with large number of simulations. Maybe in the long long run, with enough bets, one will recover all the money that have lost hehe. But I wouldn’t make such bet =P

2. mchouza says:

Thanks for your comment! I almost posted the “answer” post yesterday, but I decided to clean up the text a bit, as it was somewhat confusing.

But, correct me if I’m wrong, it would seem that the random variable M = 2 ^ N, the money gained each bet, does not have an expected value.

You are right, it has an “infinite expected value”. In the next post I will analyse why this doesn’t show in simulations.

3. […] October 17, 2010 at 10:18 pm (matemática, math, probabilidad, probability) (Follow-up to Expected value.) […]