Question

50 couples are assigned to a round table randomly. Let X be the number of couples that sits together. Calculate the expectation and the variance of X, and provide explanation for each steps.

Answer 1

to compute P(X=1) :

assume that husband 1 is seated first on table then of remaining 99 seats which are available to wife 1 only two will lead to sitting together

so, P(X=1) = 2/99

therefore, E(N) = 50.P(X=1) = 50*2/99 = 100/99

to compute the variance of N we use

Var(N) = E(N2) - E(N)2

E(N2) = 50E(x=1) + 50(50-1)E(X=1 X=2)

therfore E(N2) = 50*2/99 + 50*49*(1/49)*(2/99) = 2.02

therfore,

Var(N) = 2.02 - (100/99)2 = 0.99

P.S. (please upvote if you find the answer satisfactory for the effort)