300 hundred people apply for three jobs. 120 of the applicants are women.
(a) If three persons are selected at random, what is the probability that two are women?
(B)If three persons are selected at random, what is the probability that one is a woman?
(c) If three persons are selected at random, what is the probability that none are women?
(Round the answer to six decimal places.)
(a) (120/300) (119/299) (180/298) = ?
Use similar method for remaining problems.
To solve this problem, we need to determine the total number of possible outcomes and the number of favorable outcomes for each scenario.
(a) Probability that two out of three selected persons are women:
The total number of possible outcomes when three persons are selected from a group of 300 applicants is given by the combination formula:
C(n, r) = n! / (r! * (n - r)!)
where n is the total number of applicants and r is the number of persons being selected.
In this case, n = 300 and r = 3. So, the total number of possible outcomes is:
C(300, 3) = 300! / (3! * (300 - 3)!) = 300! / (3! * 297!) = 44,850,000
Now let's determine the number of favorable outcomes, i.e., the number of ways to select two women out of 120 women:
C(120, 2) = 120! / (2! * (120 - 2)!) = 120! / (2! * 118!) = 7,140
The probability can then be calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
P(two women) = 7,140 / 44,850,000 ≈ 0.000159
(b) Probability that one out of three selected persons is a woman:
Similarly, the number of favorable outcomes for this scenario can be calculated by determining the number of ways to select one woman out of 120 women:
C(120, 1) = 120! / (1! * (120 - 1)!) = 120! / (1! * 119!) = 120
The probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
P(one woman) = 120 / 44,850,000 ≈ 0.000003
(c) Probability that none of the three selected persons is a woman:
Again, the number of favorable outcomes can be calculated by determining the number of ways to select three persons out of 180 men:
C(180, 3) = 180! / (3! * (180 - 3)!) = 180! / (3! * 177!) = 1,026,820
The probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
P(no women) = 1,026,820 / 44,850,000 ≈ 0.023
Therefore, the rounded probabilities are:
(a) P(two women) ≈ 0.000159
(b) P(one woman) ≈ 0.000003
(c) P(no women) ≈ 0.023