Question 1101679: After being rejected for employment, Kim Kelly learns that the Bellevue Credit Company has hired only three 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 three 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 three) =
Incorrect
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! since the pool is quite large and the probability of hiring a qualified woman is the same as hiring a qualified man, then:
p = .5
q = .5
n = 22
x = 0 to 3
this is a binomial probability type problem.
the standard form is p(x) = p^x * q^(n-1) * c(n,x)
for x = 0 to 3, the formula becomes:
p(0) = .5^0 * .5^22 * c(22,0)
p(1) = .5^1 * .5^21 * c(22,1)
p(2) = .5^2 * .5^20 * c(22,2)
p(3) = .5^3 * .5^19 * c(22,3)
results are:
sum of p(x) for x equals 0 to 3 is equal to 4.27723 * 10^-4 rounded to 9 decimal places.
the decimal equivalent to this is .000427723 rounded to 9 decimal places.
rounded to 8 decimal places, this is equal to .00042772.
c(n,x) is the number of ways you can draw a set of x elements out of a set of n elements.
the formula is c(n,x) = n! / (x! * (n-x)!)
for example:
c(22,3) is equal to 22! / 3! * 19!)
this is equal to (22 * 21 * 20 * 19!) / (3! * 19!)
the 19! in the numerator and denominator cancel out to get:
(22 * 21 * 20) / (3!)
since 3! is equal to 3*2*1 = 6, this then becomes equal to:
(22 * 21 * 20) / 6
the end result is that c(22,3) is equal to 1540
p(3) is therefore equal to .5^3 * .5^19 * 1540 which is equal to 0.000367165 rounded to 9 decimal places, as shown in the attached excel spreadsheet printout.
the probability of getting less than or equal to 3 women is exceedingly small.
it is much smaller than .5 which one would expect, given that women are equally as qualified as men.
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