# A committee of three is to be selected at random from five married couples.What is the probability that no two members of the committee are married to each other?

Jul 12, 2018

$\frac{2}{5}$

#### Explanation:

The first selection is free, since there is only one person involved.

So far, one person is chosen, and five people remain: the husband/wife of the first selected, and the other two couples.

For the second selection, you have to choose among five remaining people, and you can't choose the one married to the first person. This means that you have a chanche of $\frac{4}{5}$ to pick someone not married to the first person.

So far, two people are chosen (not married to each other), and four people remain: the husband/wife of the first selected, the husband/wife of the second selected, and the last couple.

For the third selection, you can only choose anyone from the third couple. So, you have two good picks over four possible people, and $\frac{1}{2}$ chanche of succeding.

So, the "experiment" is successful if both picks go well, which has a chanche of $\frac{4}{5} \cdot \frac{1}{2} = \frac{4}{10} = \frac{2}{5}$