# Probability

## Mathematics: Standard Multiple Choice

Remember to take the time to read carefully, to note the bottom line, to choose the fastest way to solve the problem, and to match your solution to the answer choices.

Of 5 employees, 3 are to be assigned an office and 2 are to be assigned a cubicle. If 3 of the employees are men and 2 are women, and if those assigned to the office are to be chosen at random, what is the probability that the offices will be assigned to 2 of the men and 1 of the women?

First, read carefully. There is a trap lurking here; it is easy to assume that the 3 men will get the offices and the 2 women will get cubicles just because the numbers match up. However, the question says that the offices are assigned randomly, not based on gender, so you will need a more sophisticated way to solve the problem. What is the bottom line? The question asks for "the probability that the offices will be assigned to 2 of the men and 1 of the women," so abbreviate that and put it at the top of your scratch work.

prob m,m,w=?

Probabilities are simply fractions with the total number of possibilities in the denominator and the relevant possibilities in the numerator.

To find the denominator, you need the number of combinations of 3 employees that you can create from a larger group of 3 men and 2 women. To do that, draw a blank for each employee that you select from the larger group and fill in each blank with the number of possibilities it represents.

5      4      3

Multiply these numbers to get the number of ways to arrange three employees. Since you do not need to line them up in a certain order, some of these arrangements will be duplicates of the same combination. That means you will need to divide. When looking for combinations, divide by the factorial of the number of items to eliminate double-counting. In this case, there are 3 blanks, so you need to divide by 3!

$\frac{5\ast 4\ast 3}{3\ast 2}$

Canceling the 3's and the 2 will leave you with  $5\ast 2=10$  total possible combinations of 3 employees, regardless of gender. That is your denominator.

Next, you need to find the numerator. Draw three blanks again, but this time you will be more particular about what goes into them. Two blanks will be for men, and the third is for a woman. You start with three men and two women, so your blanks should look like this:

3      2      2

This time you only have two places that might get mixed up, the two blanks for male employees. multiply those blanks together and then divide by 2! to make sure you are counting combinations, not permutations. You also need to multiply by 2 to account for your female employees.

$\frac{3\ast 2}{2}\ast 2$

The 2s cancel out, leaving $3\ast 2=6$ possible combinations of 2 male and 1 female employee. This number represents the number of relevant possibilities, the numerator in the probability formula.

Put the probability formula together now that you have found both parts, and reduce it to lowest terms.

$\frac{6}{10}=\frac{3}{5}$

Finally, find this solution among the answer choices and mark it.

$A)\frac{1}{3}$

$B)\frac{2}{5}$

$C)\frac{1}{2}$

$D)\frac{3}{5}$

$E)\frac{2}{3}$