how can I know all the probabilities

Question

Five applicants are to be scheduled for job interviews at five given times. On an application form, each applicant was asked to mark two out of the five times. What is the probability that each of them can be interviewed at one of the two times they have marked?

Answer 1

Couple of assumptions here. First, assume that the interviewer is picking the timeslot randomly; that is, the interviewee has an equal chance of getting each of the timeslots (and their indication of a preferred timeslot doesn't really affect the outcome). Second, assume that the interviewees are not being offered the same timeslots, such that the outcome of one interviewee does not affect that of another; in other words, they're independent events. I think these assumptions are safe since they did not give us additional information to solve the problem if the assumptions were in fact wrong.

So let's consider one interviewee. Since there are five total timeslots and two are preferred, and each of them is equally likely to be picked, it would be a 2/5 = 0.4 probability (or 40%) that one interviewee gets a preferred timeslot. This is true of each interviewee. Each one has a 40% chance of getting one of their preferred timeslots.

Here, we are using the AND operator to join together independent events. The probability of A and B both occuring, if A and B are independent events, is P(A) * P(B). In this case, we have five independent events, each with a probability of 0.4. So it would be...

P(all five getting preferred timeslot) = 0.4 * 0.4 * 0.4 * 0.4 * 0.4 = 0.4^5 = 0.01024

That is, there is about a 1% chance that everyone gets a preferred timeslot.