Three men and three women are waiting to be interviewed for jobs. If they are all selected in random order find the probability of each event. All the women will be interviewed first. No man will be interviewed until at least two women have been interviewed.
To find the probability of each event, let's consider the possible outcomes and count the favorable outcomes for each event.
We have three men (M1, M2, M3) and three women (W1, W2, W3) waiting to be interviewed.
Event 1: All the women will be interviewed first.
Since all the women need to be interviewed first, the possible orderings of the six people would be WWWMWM. The favorable outcome is only one ordering, so the probability of this event is 1 out of the total number of possible orderings.
Total Possible Orderings:
We can calculate the total possible orderings using the concept of permutations. Since all six individuals are unique, we can arrange them in 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720 ways.
P(Event 1) = 1/720
Event 2: No man will be interviewed until at least two women have been interviewed.
For this event to occur, we need to consider two cases:
Case 1: The first two interviews consist of women only.
The possible orderings of the first two interviews would be WW, and the remaining orderings would be MMWMW. So, the favorable outcomes in this case are the total orderings of M1, M2, M3 and W3, W2, W1 (which is 3! or 3 * 2 * 1). Therefore, the probability of this case is (3!/720) or 1/120.
Case 2: The first three interviews consist of two women and one man.
The possible orderings of the first three interviews would be WWM, and the remaining orderings would be MWWMW. So, the favorable outcomes in this case are the total orderings of M1, M2, M3 and W3, W2, W1 (which is again 3!). Therefore, the probability of this case is (3!/720) or 1/120.
Since these two cases are mutually exclusive, we can sum their probabilities to get the probability of Event 2.
P(Event 2) = (1/120) + (1/120) = 2/120 = 1/60
In summary:
- The probability of Event 1 (All the women will be interviewed first) is 1/720.
- The probability of Event 2 (No man will be interviewed until at least two women have been interviewed) is 1/60.