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 exactly 20 of the 30 individuals are men, we need to use the concept of binomial probability. The binomial probability formula is:
P(X = x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(X = x) is the probability of getting exactly x successes (in this case, 20 men)
n is the total number of trials (in this case, 30 individuals)
x is the number of successful outcomes (in this case, 20 men)
nCx is the number of combinations of n things taken x at a time (this can be calculated using factorials)
p is the probability of success in a single trial (in this case, the probability of selecting a male, which is 60% or 0.6)
(1-p) is the probability of failure in a single trial (in this case, the probability of selecting a female, which is 40% or 0.4)
Let's plug in the values into the formula:
P(X = 20) = (30C20) * (0.6^20) * (0.4^(30-20))
To calculate (30C20), we need to calculate the factorial of 30 and divide it by the product of the factorial of 20 and the factorial of (30-20):
(30C20) = 30! / (20! * (30-20)!)
30! = 30 * 29 * 28 * ... * 3 * 2 * 1
20! = 20 * 19 * 18 * ... * 3 * 2 * 1
(30-20)! = 10!
P(X = 20) = (30! / (20! * 10!)) * (0.6^20) * (0.4^10)
Calculating this expression will give you the probability that exactly 20 out of the 30 individuals in the mailing list are men.