A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?
A) 1/14
B) 1/7
C) 2/7
D) 3/7
E) 1/2
When a probability problem uses the word
exactly, we can apply the following reasoning:
P(exactly n) = P(one way) * all possible ways.
Let W = woman and M = man.
P(one way):
One way to get 2 women and 2 men is WWMM.
P(1st person is W) = 5/8. (8 people, 5 of them W.)
P(2nd person is W) = 4/7. (7 people left, 4 of them W.)
P(3rd person is M) = 3/6. (6 people left, 3 of them M.)
P(4th person is M) = 2/5. (5 people left, 2 of them M.)
Since we want all of these events to happen together, we multiply the fractions:
P(WWMM) = 5/8 * 4/7 * 3/6 * 2/5 = 1/14.
All possible ways:
Any arrangement of WWMM will yield exactly 2 W and 2 M.
Thus, the result above must be multiplied by the number of ways to ARRANGE the 4 elements WWMM.
The number of ways to arrange 4 elements = 4!.
But WWMM includes IDENTICAL elements.
When an arrangement includes identical elements, we must divide by the number of ways each set of identical elements can be arranged.
The reason:
When the identical elements swap places, the arrangement DOESN'T CHANGE.
Here, we must divide by 2! to account for the two identical W's and by another 2! to account for the two identical M's:
4!/(2!2!) = 6.
Thus:
P(exactly 2 W) = 1/14 * 6 = 3/7.
The correct answer is
D.
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