hundred people apply for three jobs. 110 of the applicants are women.
(a) If three persons are selected at random, what is the probability that two are women? (Round the answer to six decimal places.)
Please proofread and clarify.
no idea
To calculate the probability that two out of three selected persons are women, we need to find two values: the total number of ways to select three persons out of 100, and the total number of ways to select two women out of 110.
Let's calculate these values step by step.
The total number of ways to select three persons out of 100 can be determined using the combination formula, which is represented as:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items, and r is the number of items being chosen. In this case, n = 100 and r = 3.
C(100, 3) = 100! / (3!(100-3)!)
= 100! / (3!97!)
Calculating the factorials:
100! = 100 × 99 × 98 × ... × 3 × 2 × 1
3! = 3 × 2 × 1
97! = 97 × 96 × ... × 3 × 2 × 1
Plugging in the values:
C(100, 3) = (100 × 99 × ... × 3 × 2 × 1) / [(3 × 2 × 1) × (97 × 96 × ... × 3 × 2 × 1)]
Simplifying the expression, we can cancel out some terms:
C(100, 3) = (100 × 99 × 98) / [(3 × 2 × 1)]
C(100, 3) = 161,700
The total number of ways to select two women out of 110 can be calculated using the combination formula again, with n = 110 and r = 2:
C(110, 2) = 110! / (2!(110-2)!)
= 110! / (2!108!)
Similarly, simplifying the expression, we have:
C(110, 2) = (110 × 109) / [(2 × 1)]
C(110, 2) = 5,995
To calculate the probability, we divide the total number of ways to select two women out of three persons by the total number of ways to select three persons:
Probability = C(110, 2) / C(100, 3)
= 5,995 / 161,700
≈ 0.037
Therefore, the probability that two out of three randomly selected persons are women is approximately 0.037.