[EMAIL PROTECTED] (Xiao Li) wrote in message
news:<[EMAIL PROTECTED]>...
> A theater has a set of seats consisting of four rows and twelve
> columns for a total of 48 seats. 10 people walk into the theater and
> each person takes a seat at random. Let X be the number of people
> with someone sitting directly in front of them. What is the expected
> value of X?
E(X) = 12*E(Y), where Y is the number of people in any given column
that have someone sitting directly in front of them.
E(Y) = Sum{E(Y|n)*P(n), n = 0...4},
where n is the number of people in the column,
P(n) = the probability of getting n people in the column
= C(4,n)*C(44,10-n)/C(48,10), where C(a,b) = a!/(b!(a-b)!),
and E(Y|n) = k(n)/C(4,n), where k(n) is the total number of people
with someone in front of them, summed over all C(4,n) configurations
of n people in the 4-seat column. k(n) is most conveniently obtained
by inspection of the configurations, giving:
n 0 1 2 3 4
k(n) 0 0 3 6 3
So E(X) = 12*Sum{k(n)*C(44,10-n), n = 2...4}/C(48,10)
= 12*558900/4669920
= 1.436
.
.
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