Question

1. when students register for a popular course at a particular university, one of three things can happen. they are either (1) enrolled in the course, (2) put on a wait list, or (3) denied enrollment. based on your understanding of probability models, which one of the following statements must be true?

a. the probabilities of being enrolled in the course or being put on the wait list should sum to 1.
b. the probability of being denied enrollment in the course is equal to 1 minus the probability of being enrolled in the course.
c. if we consider being denied enrollment as basically a loss of enrollment, the probability of being denied enrollment should be negative.
d. we can find the probability of being put on the wait list by averaging the probabilities of being enrolled in the course and being denied enrollment in the course.
e. none of the above statements is true.

Explanation:

Step1: Analyze Option A

The three events (enrolled, wait - list, denied) are mutually exclusive and exhaustive. The sum of probabilities of all three events should be \(1\), not just enrolled and wait - list. So, \(P(\text{enrolled})+P(\text{wait - list})
eq1\).

Step2: Analyze Option B

Since \(P(\text{enrolled})+P(\text{wait - list})+P(\text{denied}) = 1\), then \(P(\text{denied})=1-(P(\text{enrolled}) + P(\text{wait - list}))
eq1 - P(\text{enrolled})\)

Step3: Analyze Option C

Probability of an event \(E\) satisfies \(0\leq P(E)\leq1\). Probability of being denied enrollment cannot be negative.

Step4: Analyze Option D

There is no such rule that \(P(\text{wait - list})=\frac{P(\text{enrolled})+P(\text{denied})}{2}\)

Answer:

E. None of the above statements is true.