#### 1. University admissions.

Suppose a university announced that it admitted 2,500 students for the following year’s freshman class. However, the university has dorm room spots for only 1,786 freshman students. If there is a 70% chance that an admitted student will decide to accept the offer and attend this university, what is the approximate probability that the university will not have enough dormitory room spots for the freshman class?

## Solution.

Want to find the probability that there will be 1,786 or more enrollees. Using the normal approximation, with \(\mu= np = 2,500 \times 0.7 = 1750\) and \(\sigma = \sqrt{np(1-p} = \sqrt{2,500 \times 0.7 \times 0.3} \approx 23\text{,}\) \(Z = 1.61\text{,}\) and \(P(Z > 1.61) = 0.0537\text{.}\) With a 0.5 correction: 0.0559.