Question

The probability that a freshman entering a university will graduate after a 4-year period is 0.64. If 8 freshmen are selected at random, find the probability that at least 3 of them will graduate after a 4-year period

Answer 1

This is a binomial probability problem with n=8, p=0.64, and we want to find the probability of at least 3 successes (graduating) out of 8 trials (freshmen).

One way to solve this is to use the complement rule: the probability of at least 3 successes is equal to 1 minus the probability of 0, 1, or 2 successes.

P(at least 3 successes) = 1 - P(0 or 1 or 2 successes)

Using the binomial probability formula, we can calculate the probability of each of the three cases:

P(0 successes) = C(8, 0) * 0.64^0 * (1-0.64)^8 = 0.0002

P(1 success) = C(8, 1) * 0.64^1 * (1-0.64)^7 = 0.0037

P(2 successes) = C(8, 2) * 0.64^2 * (1-0.64)^6 = 0.0286

Therefore,

P(at least 3 successes) = 1 - (0.0002 + 0.0037 + 0.0286) = 0.9675

So the probability that at least 3 out of 8 freshmen will graduate after a 4-year period is approximately 0.9675 or 96.75%.