Question

1) Suppose that the 10 people of 5 married couples (with each couple consisting of one husband and one wife) will be randomly seated at a round table having 10 chairs. Letting X be the number of wives who will be seated next to their husband, give the value of E(X).

Answer 1

Suppose that the 10 people of 5 married couples (with each couple consisting of one husband and one wife) will be randomly seated at a round table having 10 chairs. Letting X be the number of wives who will be seated next to their husband

Label the wives 1 through 5 and

let X_i = 1 if the i^th wife is seated next to her husband and X_i = 0 otherwise.

be the number of wives seated next to their husbands. Then

Given where the i^th wife sits, her husband may sit in one of the 9 remaining seats, two of which are next to his wife. So P{Xi = 1} = 2/9 and hence
hence required value of E[X] = 10/9.

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