The mailing list of an agency that markets scuba-diving trips to the Florida Keys contains 60% males and 40% females. What is the probability that 20 of the 30 are men?
To find the probability that 20 out of 30 people are men, given that the mailing list contains 60% males and 40% females, we need to use the concept of binomial probability.
The binomial probability formula is given by:
P(X = k) = (n C k) * p^k * q^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes
- n is the total number of trials (in this case, the total number of people)
- k is the number of successes (in this case, the number of men)
- p is the probability of success (in this case, the probability of a person being male)
- q is the probability of failure (in this case, the probability of a person being female)
In this problem, n = 30 (total number of people) and we want to find the probability of having 20 men (k = 20). The probability of a person being male (p) is 60% or 0.6, and the probability of a person being female (q) is 40% or 0.4.
Using these values, we can calculate the probability as follows:
P(X = 20) = (30 C 20) * (0.6^20) * (0.4^10)
The binomial coefficient (n C k) represents the number of ways to choose k successes from n trials and is calculated as:
(n C k) = n! / (k! * (n-k)!)
In this case, we have:
(30 C 20) = 30! / (20! * (30-20)!)
Now we can calculate the probability:
P(X = 20) = (30 C 20) * (0.6^20) * (0.4^10)
Calculating this expression will give us the probability that 20 out of 30 people are men on the mailing list.