Binomial Distribution

Question

After being rejected for employment, Kim Kelly learns that the Bellevue Credit Company has hired only four women among the last 22 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 four or fewer women when 22 people are hired, assuming that there is no discrimination based on gender. 

(Report answer accurate to 8 decimal places). 

P(at most four) = ?

Because this is a serious claim, we will use a stricter cutoff value for unusual events. We will use 0.5% as the cutoff value (1 in 200 chance of happening by chance). With this in mind, does the resulting probability really support such a charge? 

1. no, this does not support a charge of gender discrimination
2. yes, this supports a charge of gender discrimination

Answer 1

You could use a cumulative binomial distribution function or you could use a cumulative normal distribution approximation. Brute force, just add up the probabilities of k = 0 to 4 happening out of an n of 22:

Formula is C(n,k) p^k(1-p)^k In this case C(22, k) (1/2)²² where C(n,k) = n!/((n-k)!(k!))

P = (C(22,0)+C(22,1)+C(22,2)+C(22,3)+C(22,4))/2²²