# Probably easy expected value problem

1. Mar 20, 2007

### mkkrnfoo85

Problem:-------
Suppose n people have throat cultures, and the cultures are then completely
mixed up. If we randomly pair off the n peoples’ names with the n cultures,
what is the expected number of correct labels?
---------------

So, here's my logic on the solution:

I think there are 2 random variables here, X1 for names, X2 for the cultures.
The formula for the expected value for 2 random variables is:

$$\sum_1^n \sum_1^n h(X1,X2)*prob(X1,X2)$$ for some weight function h.

I chose h(X1,X2) = 1. Also, I said prob(X1,X2) = (1/n^2). I hope all that is right so far. So just using the expected value equation, I simply said the answer is 1.

$$\sum_1^n \sum_1^n (1)* \left \frac{1}{n^2} \right = 1$$

Correct thinking? If not, what am I doing wrong?

Mark

Last edited: Mar 20, 2007
2. Mar 20, 2007

### christianjb

N labels on N bottles.

1/N chance for label i to correspond to bottle i.

Average number of correct labels= sum_i=1,N * chance for label i
=1

So I agree.