Questions: If the probability is 0.65 that a marriage will end in divorce within 20 years, what is the probability that out of 4 couples just married, in the next 20 years (A) None will be divorced? (B) All will be divorced? (C) Exactly 3 will be divorced? (D) At least 3 will be divorced?

Solution Steps

To solve this problem, we can use the binomial probability formula. The probability of a certain number of successes in a series of independent trials can be calculated using this formula. Here, a "success" is defined as a divorce occurring.

(A) For none divorced, calculate the probability of 0 divorces. (B) For all divorced, calculate the probability of 4 divorces. (C) For exactly 3 divorced, calculate the probability of 3 divorces. (D) For at least 3 divorced, sum the probabilities of 3 and 4 divorces.

We are given the probability \( p = 0.65 \) that a marriage will end in divorce within 20 years. We need to find the probabilities for different scenarios involving 4 couples.

The binomial probability formula is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n \) is the number of trials, \( k \) is the number of successes, and \( p \) is the probability of success.

• (A) None Divorced: \( k = 0 \) \[ P(X = 0) = \binom{4}{0} (0.65)^0 (0.35)^4 = 0.0150 \]

• (B) All Divorced: \( k = 4 \) \[ P(X = 4) = \binom{4}{4} (0.65)^4 (0.35)^0 = 0.1785 \]

• (C) Exactly 3 Divorced: \( k = 3 \) \[ P(X = 3) = \binom{4}{3} (0.65)^3 (0.35)^1 = 0.3845 \]

• (D) At Least 3 Divorced: Sum of probabilities for \( k = 3 \) and \( k = 4 \) \[ P(X \geq 3) = P(X = 3) + P(X = 4) = 0.3845 + 0.1785 = 0.5630 \]

Final Answer