Three hundred people apply for three jobs. 80 of the applicants are women If three persons are selected at random, what is the probability that one is a woman? (Round the answer to six decimal places.)(d) If three persons are selected at random, what is the probability that none is a woman? (Round the answer to six decimal places.)
prob(woman) = 80/300 = 2/75
prob(not woman) = 73/75
prob(1 women of 3)
= C(3,1) (2/75)(73/75)^2
= appr .075790
prob(no woman) = (73/75)^3 = .922114
To find the probability that one person selected at random is a woman, we need to calculate the ratio of the number of favorable outcomes to the total number of outcomes.
1. Determine the number of favorable outcomes:
Since there are 80 women among the 300 applicants, we have 80 possible choices for the first position.
2. Determine the total number of outcomes:
To calculate the total number of outcomes, we need to determine the number of ways to choose 3 people out of 300, which is denoted as "300 choose 3" or written as "C(300, 3)". This can be calculated using the combination formula:
C(n, k) = n! / (k!(n-k)!)
where n! denotes n factorial, which is the product of all positive integers less than or equal to n.
Calculating C(300, 3):
C(300, 3) = 300! / (3!(300-3)!)
= 300! / (3! * 297!)
= (300 * 299 * 298 * 297!) / (3 * 2 * 1 * 297!)
= (300 * 299 * 298) / (3 * 2 * 1)
= 2,682,600
3. Calculate the probability:
The probability that one person selected at random is a woman can be calculated by dividing the number of favorable outcomes by the total number of outcomes:
Probability = Favorable Outcomes / Total Outcomes
= 80 / 2,682,600
≈ 0.000029825 (rounded to six decimal places)
Therefore, the probability that one person selected at random is a woman is approximately 0.000029825.
To find the probability that none of the three selected persons are women, we need to follow similar steps:
1. Determine the number of favorable outcomes:
Since there are 220 men among the 300 applicants (300 - 80 women), we have 220 possible choices for the first position.
2. Determine the total number of outcomes (using the same combination formula):
C(300, 3) = 2,682,600 (as calculated before)
3. Calculate the probability:
The probability that none of the three selected persons are women can be calculated by dividing the number of favorable outcomes by the total number of outcomes:
Probability = Favorable Outcomes / Total Outcomes
= 220 / 2,682,600
≈ 0.000082011 (rounded to six decimal places)
Therefore, the probability that none of the three selected persons are women is approximately 0.000082011.