Question

After being rejected for employment, Kim Kelly learns that the Bellevue Credit Company has hired only five women among the last 19 new employees. She also learns that the pool of applicants is very large, with an approximately equal number of qualified men as qualified women. Help her address the charge of gender discrimination by finding the probability of getting five or fewer women when 19 people are hired, assuming that there is no discrimination based on gender.Does the resulting probability really support such a charge?

Answer 1

Answer:

The probability that 5 or fewer women are hired, assuming no gender discrimination, is 0.0317; we can use this result to support her charge of gender discrimination.

Step-by-step explanation:

If we are assuming that the women and the men are equally qualified, then the probability for each employee that is hired the probability for it to be a women should be 1/2. Note that the fact that more men that women are hired in a sample might not be disctrimination: for example, if 2 men are hired out of 2 employees, that can happen with probability 1/4, so it is quite common. In order to support her charge for gender discrimination, we need at least a probability less that 0.05 that 5 (or less) women are hired out of 19 employees.

Since each configuration is equally probable, we will count the total amount of possible cases that 5 or less women are hired, and dividide it by the total amount of cases, 2¹⁹.

• 0 women hired: one possible case: every employee is male
• 1 women hired: 19 possible cases
• 2 women hired: ${19 \choose 2} = 171$ possible cases
• 3 women hired: ${19 \choose 3} = 969$ possible cases
• 4 women hired: ${19 \choose 4} = 3876$ possible cases
• 5 women hired: ${19 \choose 5} = 11628$ possible cases

Thus, there are a total of 11628+3876+969+171 = 16644 possible cases out of 2¹⁹ ones. All of them with seemingly equal probability. As a consequence, the probability of 5 or less women to be hired out of 19 employees, assuming that the probability to hire 1 is 1/2, is

16644/2¹⁹ = 0.0317 < 0.05

The probability that 5 or fewer women are hired, assuming no gender discrimination, is 0.0317. Since the probability is so low, we can conclude that for the employer, a woman equally qualified as a man is less likely to be hired, therefore, we can support her charge of gender discrimination.