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Probably easy expected value problem

  1. Mar 20, 2007 #1
    Hello, just a bit insecure about my answer.

    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:

    [tex]\sum_1^n \sum_1^n h(X1,X2)*prob(X1,X2)[/tex] 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.

    [tex]\sum_1^n \sum_1^n (1)* \left \frac{1}{n^2} \right = 1[/tex]

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

    Thanks in advance,

    Last edited: Mar 20, 2007
  2. jcsd
  3. Mar 20, 2007 #2
    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

    So I agree.
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