Help! Real-life probability problem. Can't figure it out!?

Question

I read a recent news article (http://hamptonroads.com/2008/10/chesapeake-police-groups-say-theyll-sue-if-city-voids-tests) that brought up a great probability problem.  I just can't figure it out.  

The police department in Chesapeake, Virginia holds a policy exam for its officers. The thirty top scorers on the exam are made eligible for promotion.  This year, the city of Chesapeake has considered invalidating the results of the exam, because all of the top thirty scorers were white males. The city has contracted an analyst to determine if the exam was biased against minorities and women.

Here's the problem... Of the 144 candidates who took the exam, 135 were white men.  Only nine of the examinees were either women or racial minorities.  

Assuming that all other things are equal, how many women or minorities should we expect to be in the top thirty, on average?  Also, what is the probability of the actual outcome (ie. that all thirty top candidates would be white men)? 

Any help at all would be appreciated, but anyone who can tell me HOW to work this kind of problem gets ten points.  

Thanks!

Anonymous · Accepted Answer

let M denote white males, and X denotes non-white males. since the proportion of X in the population is (144-135)/144 = 0.0625. So out of the 30 top scores, we would expect 30*0.0625 = 1.875 X in it. 
for the second quesiton, we have to ask what's the probability of choosing all of the top 30 scorers and found none X? we can assume that each trial is a bernoulli trail with p = 0.0625, where p is the probability of success. so X has a binomial distribution, and we have to find P( X = 0), using Maple, I am able to get 0.14426
this means there is a 14.425% chance that you will find 0 non-white males in the top 30, in other words, the probability that you will find at least 1 non-white males in the top 30 is 85.57%

Nate E · Answer

examine by percentage:
135/144 to determine 93.75% are white men.

you know there are thirty participants for promotion.
so 30 * 93.75% = 28.125

based off this you should expect 2 or 1 people to be minorities and women.

susann · Answer

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Help! Real-life probability problem. Can't figure it out!?

3 Answers