Four 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 in the following statements (a) All the women will be interviewed first. (b) All the men will be interviewed first (c) The first person interviewed will be a women (d) The second person interviewed will be a women (e) The last person interviewed will be a women
(a) 3/7 * 2/6 * 1/5
(b) 4/7 * 3/6 * 2/5
see what you can do with the others
To find the probability of each event, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event.
Total number of possible outcomes:
There are a total of 7 people waiting to be interviewed, so there are 7! (7 factorial) ways to arrange them.
a) All the women will be interviewed first:
Favorable outcomes: There are 3 women who can be interviewed first. Once the women are interviewed, the remaining 4 men can be arranged among themselves in 4! ways.
Number of favorable outcomes = 3! * 4! = 6 * 24 = 144
Probability = Number of favorable outcomes / Total number of possible outcomes = 144 / 7! = 144 / 5040 ≈ 0.0286 or 2.86%
b) All the men will be interviewed first:
Favorable outcomes: There are 4 men who can be interviewed first. Once the men are interviewed, the remaining 3 women can be arranged among themselves in 3! ways.
Number of favorable outcomes = 4! * 3! = 24 * 6 = 144
Probability = Number of favorable outcomes / Total number of possible outcomes = 144 / 7! = 144 / 5040 ≈ 0.0286 or 2.86%
c) The first person interviewed will be a woman:
Favorable outcomes: There are 3 women who can be interviewed first. Once the first woman is interviewed, the remaining 6 people can be arranged among themselves in 6! ways.
Number of favorable outcomes = 3 * 6!
Probability = Number of favorable outcomes / Total number of possible outcomes = (3 * 6!) / 7! = 3 / 7 ≈ 0.4286 or 42.86%
d) The second person interviewed will be a woman:
Favorable outcomes: After choosing a woman for the first position, there are 3 women and 6 people remaining to be arranged. The second position can be filled by 1 of the 3 women and the remaining people can be arranged in 6! ways.
Number of favorable outcomes = 3 * 6!
Probability = Number of favorable outcomes / Total number of possible outcomes = (3 * 6!) / 7! = 3 / 7 ≈ 0.4286 or 42.86%
e) The last person interviewed will be a woman:
Favorable outcomes: There are 3 women who can be interviewed last. Once the last woman is chosen, the remaining 6 people can be arranged in 6! ways.
Number of favorable outcomes = 3 * 6!
Probability = Number of favorable outcomes / Total number of possible outcomes = (3 * 6!) / 7! = 3 / 7 ≈ 0.4286 or 42.86%
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.
Given information:
Number of men (M) = 4
Number of women (W) = 3
(a) All the women will be interviewed first:
First, let's determine the total number of possible outcomes.
There are 7 people to be interviewed, so there are 7! (factorial) ways to arrange them.
Next, let's determine the number of favorable outcomes.
Since we want all the women to be interviewed first, we can arrange the 3 women among themselves in 3! ways. After that, we can arrange the 4 men among themselves in 4! ways. So the number of favorable outcomes is 3! * 4!.
The probability of this event is:
P(all women first) = favorable outcomes / total outcomes
P(all women first) = (3! * 4!) / 7!
(b) All the men will be interviewed first:
Using the same logic as in part (a), we can determine the number of favorable outcomes for this event.
We arrange the 4 men among themselves in 4! ways and then arrange the 3 women among themselves in 3! ways.
So the number of favorable outcomes is 4! * 3!.
The probability of this event is:
P(all men first) = favorable outcomes / total outcomes
P(all men first) = (4! * 3!) / 7!
(c) The first person interviewed will be a woman:
The first person can be any of the 3 women, and the remaining 6 people can be arranged in 6! ways.
The probability of this event is:
P(first person is a woman) = favorable outcomes / total outcomes
P(first person is a woman) = 3 * 6! / 7!
(d) The second person interviewed will be a woman:
The first person can be any of the 7 people, and the second person must be a woman.
There are 3 women to choose from, and then the remaining 5 people can be arranged in 5! ways.
The probability of this event is:
P(second person is a woman) = favorable outcomes / total outcomes
P(second person is a woman) = 7 * 3 * 5! / 7!
(e) The last person interviewed will be a woman:
The last person must be a woman, so there is only 1 choice for the last person.
The remaining 6 people can be arranged in 6! ways.
The probability of this event is:
P(last person is a woman) = favorable outcomes / total outcomes
P(last person is a woman) = 1 * 6! / 7!
Now you have the formulas to calculate the probabilities for each event using the information provided.