Final answer:
The probability that exactly two out of eight randomly chosen students are female is approximately 0.26844.
Step-by-step explanation:
To find the probability that exactly two out of eight randomly selected students are female when 60% of the university's student body is female, we can use the binomial probability formula:
P(X=k) = C(n, k) * p^k * q^(n-k), where:
P(X=k) is the probability of getting exactly k successes,
C(n, k) is the number of combinations of n items taken k at a time,
p is the probability of success (60% or 0.6),
q is the probability of failure (40% or 0.4),
n is the number of trials (8),
k is the number of successful outcomes we want (2).
Plugging the given values into the formula, we have:
P(X=2) = C(8, 2) * (0.6)^2 * (0.4)^(8-2).
Simplifying the calculation, we get:
P(X=2) = 28 * 0.36 * 0.262144 = 0.26844.
Therefore, the probability that exactly two of the eight randomly selected students are female is approximately 0.26844.