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. No man will be interviewed until at least two women have been interviewed.
To find the probability of each event, we need to calculate the total number of possible outcomes and the number of favorable outcomes for each event.
Let's start by calculating the total number of possible outcomes.
Since there are six people in total (three men and three women), there are 6! (6 factorial) ways to arrange them in random order. Factorial means multiplying a number by all the positive integers less than it. So, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.
Now let's calculate the number of favorable outcomes for each event.
Event 1: No man will be interviewed until at least two women have been interviewed.
For this event to occur, the first two interviews must be conducted with women. There are three women available, so there are 3! (3 factorial) ways to arrange their interviews.
After the first two women have been interviewed, we have three remaining people (one man and two women) for the last interview. Again, there are 3! ways to arrange them.
Therefore, the number of favorable outcomes for event 1 is 3! × 3!.
Event 2: All three women are interviewed before any man.
For this event to occur, the three women must be interviewed first. There are 3! ways to arrange their interviews.
After the women have been interviewed, there are three men remaining. So, there are 3! ways to arrange their interviews.
Therefore, the number of favorable outcomes for event 2 is 3! × 3!.
Now, we can calculate the probability of each event.
Probability of Event 1 = (Number of favorable outcomes for event 1) / (Total number of possible outcomes)
Probability of Event 1 = (3! × 3!) / 6! = (3 × 2 × 1 × 3 × 2 × 1) / (6 × 5 × 4 × 3 × 2 × 1)
= 36 / 720
= 1/20
Probability of Event 2 = (Number of favorable outcomes for event 2) / (Total number of possible outcomes)
Probability of Event 2 = (3! × 3!) / 6! = (3 × 2 × 1 × 3 × 2 × 1) / (6 × 5 × 4 × 3 × 2 × 1)
= 36 / 720
= 1/20
Therefore, the probability of Event 1 and Event 2 is both 1/20 or 0.05.