Final answer:
The probability that 15 couples must be asked to enlist 5 couples for a new childbirth program, given a 0.2 chance of yes from any one couple, is 5.4% or option B) 0.054.
Step-by-step explanation:
The question provided asks about the probability that 15 couples must be asked to recruit 5 couples for participation in a new natural childbirth regimen, given that the probability of a single couple agreeing to participate is 0.2. This is a problem that can be solved using the negative binomial distribution because it deals with a sequence of independent trials until a fixed number of successes are observed.
To find the probability that exactly 15 couples must be asked before 5 agree to participate, we are looking at the scenario where the first 4 couples agree and the 5th couple is the 15th asked, which means the last 10 couples asked (from the 5th to the 14th) did not agree to participate. This is calculated as:
Probability = Combination(14, 4) x (0.2)^5 x (0.8)^10
Where combination(14, 4) is the number of ways to choose which 4 of the first 14 couples asked will say yes, (0.2)^5 is the probability of getting 5 yeses, and (0.8)^10 is the probability of getting 10 noes. Calculating this gives:
Probability = 1001 x (0.2)^5 x (0.8)^10 ≈ 0.054
Therefore, the correct answer is B) 0.054 (5.4%).