Question

Ten married couples are seated randomly in a circle. Find the expected number of wives sitting next to their husbands. Hint: Let X be the r., the number of wives seated next to their husband

Let W ₁ = wife i ; i=1,2,...10
Let X ₁ If W ₁ is sitting next to her husband for i=1,2,...10 and X ₁=0 if W₁ is NOT sitting next to her husband for is i=1,2,...10
Then X=X₁ + X ₂ + +X ₂ ₁₀.

Answer 1

To find the expected number of wives sitting next to their husbands, calculate the expected value of X, where X represents the number of wives sitting next to their husbands.

Step-by-step explanation:

Expected number of wives sitting next to their husbands

To find the expected number of wives sitting next to their husbands, we need to calculate the expected value of X, where X represents the number of wives sitting next to their husbands.

1. Let X₁ be the indicator variable for wife ₁ sitting next to her husband. X₁ = 1 if wife ₁ is sitting next to her husband, X₁ = 0 otherwise.
2. Similarly, define X₂, X₃, ..., X₁₀ for the other wives.
3. The total number of wives sitting next to their husbands is given by X = X₁ + X₂ + X₃ + ... + X₁₀.

To calculate the expected value of X, we need to sum up the expected values of X₁, X₂, X₃, ..., X₁₀. Since each wife's seating arrangement is independent, the expected value of each Xᵢ is the probability that wife i is sitting next to her husband.

Thus, the expected number of wives sitting next to their husbands is E(X) = E(X₁) + E(X₂) + E(X₃) + ... + E(X₁₀).