Sanderson M. Smith
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Assume you are a lawyer facing the following fictional situation:
A company has 25 employees, 6 of whom are minorities.
Five employees are randomly selected to receive a bonus of $10,000.
The selection is maid, and the selected group of 5 contains no minorities.
The minority group suspects discrimination. They come to you with the thought of taking legal action against the company.
Would you, as a lawyer, take this case?
Does the group have reason to suspect discrimination in the selection process?
Is there reason to think that the group of 5 was not randomly selected?
This presents an excellent opportunity for to use simulation to see how unusual it would be to obtain a random sample of 5 with no minorities. Thinking ahead, if you come to the conclusion that discrimination has been taken place, you (the lawyer) could put 19 red marbles and 6 blue marbles in a bag, thoroughly mix them up, and have a jury member reach in and pick a sample of 5 marbles. This could be done many times, recording the number of blue (minority) marbles in each sample. The jury could record how many times a sample with no minorities was produced. This would give them some insight into the likelihood of such an event occurring through random selection.
However, it would be wise to use math power and attempt to determine what we can "expect" if we perform the simulation described. After all, there is little sense in pursuing a lawsuit with ample justification for the discrimination claim.
Let x be the number of possible minorities in the selected group. Then x is a random variable that can assume the following values: 0, 1, 2, 3, 4, and 5. If the selection process is random, then the probabilities associated with each value of x are indicated in the following table:
x = number of minorities in selected group
p = probability that selected group contains x minorities
Product = xp
0.21886 = (6C0)(19C5)/(25C5)
0.43772 = (6C1)(19C4)/(25C5)
0.27357 = (6C2)(19C3)/(25C5)
0.06437 = (6C3)(19C2)/(25C5)
0.00011 = (6C5)(19C0)/(25C5)
1.0000 = 100%
As the chart indicates, with random selection, there is approximately a 22% chance of having no minorities in the group. That is, approximately 1 out of 5 groups randomly selected would be "expected" to contain no minorities. It's highly unlikely that you, as a lawyer, would be successful in proving that discrimination exists based solely on the fact that the chosen committee had no minorities.
If enough simulations are performed, you should get approximately the same results. Overall, roughly one in five simulations should produce a sample containing no minorities.
This is an excellent spreadsheet exercise. In Excel, =combin(19,3) is 19C3 = 969. A spreadsheet also allows one to produce a beautiful chart displaying the probabilities. A graphical display is often more powerful than a numeric display. Below you see a dot plot display showing the expected number of times each value of x is "expected" to occur in 100 trials. No value is displayed for x = 5 simply because one would expect to see 5 minorities in just 1 in every 8,855 random samples. Even this simple plot is effective in showing that x = 0 is not that unusual.
There are 25C5 = 53,130 possible samples of 5 people that could be chosen from a set of 25 people. If one recorded the number of minorities in each of these samples, and then found the average (mean) of the 53,130 numbers, this mean would be 1.2, as indicated in the table. That is, the expected number of minorities in a sample of five is 1.2.
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