Final answer:
The student's question is about finding the probability of having at most 2 men in a random sample of 14 people from a theatre audience where on average 34% are men. This is solved using the binomial probability formula, which involves computing the probabilities for having exactly 0, 1, and 2 men and summing them up.
Step-by-step explanation:
The question involves calculating the probability that in a random sample of 14 people from a theatre audience, at most 2 are men, given that on average, 34% of the people who go to the theatre are men. This is a binomial probability problem because we have two possible outcomes for each individual (either a man or not a man), a fixed number of trials (14 people), a fixed probability of success (34% or 0.34), and each trial is independent.
To find the probability of at most 2 men, we need to calculate the probability of getting exactly 0 men, exactly 1 man, and exactly 2 men, and then sum these probabilities up. The formula for the binomial probability of exactly k successes in n trials is:
P(X = k) = (n choose k) * (p)^k * (1-p)^(n-k)
Where:
- n = number of trials (14 people)
- k = number of successes (number of men, which we'll calculate for 0, 1, and 2)
- p = probability of success (0.34)
- (n choose k) = combination of n items taken k at a time
After calculating the probabilities for each case (k = 0, 1, 2), we add them together to find the total probability of having at most 2 men.