Question

In a large population of nurses, suppose 20% of the nurses would prefer the night shift. If a random sample of 10 nurses is taken, what is the probability less than 3 are nurses that prefer the night shift?

Answer 1

This is a binomial probability problem with n = 10, p = 0.2, and we need to find the probability that less than 3 nurses prefer the night shift.

We can use the binomial probability formula:

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

where X is the number of nurses in the sample who prefer the night shift.

Using the binomial probability formula, we can calculate:

P(X = 0) = (10 choose 0) * 0.2^0 * 0.8^10 = 0.1074
P(X = 1) = (10 choose 1) * 0.2^1 * 0.8^9 = 0.2684
P(X = 2) = (10 choose 2) * 0.2^2 * 0.8^8 = 0.3019

Therefore,

P(X < 3) = 0.1074 + 0.2684 + 0.3019 = 0.6777

The probability that less than 3 nurses in the sample prefer the night shift is approximately 0.6777 or 67.77%.