Question

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?
the probabilities of being enrolled in the course or being put on the wait list should sum to 1.
the probability of being denied enrollment in the course is equal to 1 minus the probability of being enrolled in the course.
if we consider being denied enrollment as basically a loss of enrollment, the probability of being denied enrollment should be negative.
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.
none of the above statements is true.

Explanation:

Step1: Analyze the first option

Let \(P(E)\) be the probability of being enrolled, \(P(W)\) be the probability of being on the wait - list, and \(P(D)\) be the probability of being denied. By the law of total probability \(P(E)+P(W)+P(D) = 1\). The first option says \(P(E)+P(W)=1\), which is only true if \(P(D) = 0\), not in general.

Step2: Analyze the second option

The second option says \(P(D)=1 - P(E)\). But from \(P(E)+P(W)+P(D)=1\), we have \(P(D)=1-(P(E) + P(W))\), not \(1 - P(E)\) unless \(P(W)=0\).

Step3: Analyze the third option

Probabilities are non - negative (\(0\leq P(X)\leq1\) for any event \(X\)). So the probability of being denied enrollment \(P(D)\geq0\). The statement that \(P(D)\) is negative is false.

Step4: Analyze the fourth option

There is no basis for the formula \(P(W)=\frac{P(E)+P(D)}{2}\). Probabilities of mutually exclusive and exhaustive events follow \(P(E)+P(W)+P(D) = 1\), not an averaging rule.

Answer:

None of the above statements is true.