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# Using set theory to solve for the probability

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A survey of a group of military personnel revealed the following information:
95 officers
90 minorities
90 women
40 women officers
38 minority women
40 minority officers
23 women minority officers
16 caucasian male enlisted personnel

How many personnel were:
a. interviewed
b. enlisted minority women
c. male minority officers
d. enlisted
e. officers, but not women or minority
f. enlisted women
g. minority women, but not officers

##### Solution Summary

Set theory is used to solve for various probabilities. This is a well presented solution.

##### Solution Preview

We use set theory to solve the problem.
Suppose P is the set of all personnel interviewed. M is the set of all males. W is the set of all women. O is the set of all officers. N is the set of all minorities.
From the condition, we have:
95 officers, this means |O|=95
90 minorities, this means |N|=90
90 women, this means |W|=90
40 women officers, this means |W and O|=40
38 ...

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