Final answer:
To find the expected number of wives sitting next to their husbands, define X as the random variable for this event and estimate using probabilities. Each wife has a 2/19 chance of sitting next to her husband, and with ten wives, the expected number is E(X) = 10 * (2/19).
Step-by-step explanation:
The question asks to find the expected number of wives sitting next to their husbands when ten married couples are seated randomly in a circle.
To solve this problem, we take a probabilistic approach by defining X as the random variable representing the number of wives seated next to their husband. For each wife, we can deem her to be sitting next to her husband as a success with probability p. Since there are ten wives, we can view this as ten independent Bernoulli trials.
In a circle of twenty seats, each wife has two possible seats next to her husband, and since the seating is random, the probability p that any given wife is sitting next to her husband is 2/19 (there are 19 seats other than her own, and two of them are next to her husband).
The expected value of X, which is the expected number of wives sitting next to their husbands, is calculated by summing up the individual probabilities for each of the ten wives. Thus, the expected value E(X) would be 10 times p, giving us an expected number of E(X) = 10 * (2/19).