At a large university it is known that 30% of the students eat at dinning halls on campus. the director of student life is going to take a random sample of 300 students. what is the probability that less than a quarter of the sampled students eat at dinning halls on campus?

Question: At a large university it is known that 30% of the students eat at dinning halls on campus. the director of student life is going to take a random sample of 300 students. what is the probability that less than a quarter of the sampled students eat at dinning halls on campus?

This is a question that can be solved using the binomial distribution. The binomial distribution describes the probability of obtaining k successes in n binomial experiments, where each trial has only two possible outcomes: success or failure¹. In this case, the success is defined as a student eating at dining halls on campus, and the failure is defined as a student not eating at dining halls on campus. The probability of success is given as p = 0.3, and the probability of failure is given as q = 1 - p = 0.7. The number of trials is n = 300, and the number of successes is k = 0.25  n = 75. The probability that X = k successes can be found by the following formula:

$$P(X=k) = \binom{n}{k}p^kq^{n-k}$$

where $\binom{n}{k}$ is the binomial coefficient that represents the number of ways to choose k items from n items. To find the probability that less than a quarter of the sampled students eat at dining halls on campus, we need to sum up the probabilities for all possible values of k from 0 to 75. This can be done using a calculator or a software program, such as Excel or R. The result is approximately 0.049, which means that there is about a 4.9% chance that less than a quarter of the sampled students eat at dining halls on campus.