After being rejected for employment, Kim Kelly learns that the Bellevue Credit Company has hired only two women among the last 18 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 two or fewer women when 18 people are hired, assuming that there is no discrimination based on gender.
(Report answer accurate to 8 decimal places).
P(at most two) =
To calculate the probability of getting two or fewer women when 18 people are hired, assuming no gender discrimination, we can use the concept of a binomial distribution.
In this case, let's define the following variables:
- n = number of trials (in this case, the number of people hired) = 18
- p = probability of success (in this case, the probability of hiring a woman) = 0.5 (since it's assumed that the pool of qualified applicants has an equal number of men and women)
- X = number of successes (in this case, the number of women hired)
The probability function for a binomial distribution is given by the formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where (n C k) represents the number of combinations of n items taken k at a time, and it can be calculated as:
(n C k) = n! / (k! * (n - k)!)
The probability of getting two or fewer women is equal to the sum of the probabilities of hiring 0, 1, or 2 women:
P(at most two) = P(X = 0) + P(X = 1) + P(X = 2)
Now, let's calculate the individual probabilities:
P(X = 0) = (18 C 0) * (0.5^0) * (0.5^(18 - 0))
P(X = 1) = (18 C 1) * (0.5^1) * (0.5^(18 - 1))
P(X = 2) = (18 C 2) * (0.5^2) * (0.5^(18 - 2))
To calculate these probabilities accurately, we need to use a calculator, spreadsheet program, or statistical software.
Once we have the individual probabilities, we can sum them up to get the final answer:
P(at most two) = P(X = 0) + P(X = 1) + P(X = 2)
Make sure to round the answer to 8 decimal places as specified in the problem.