Suppose there are eight socks in a bag, numbered one through eight, which can be grouped into four pairs: {1,2}, {3,4}, {5,6}, or {7,8}. the socks of each pair have the same color; different pairs have different colors. suppose there are four (distinct! ) people present, and one at a time, they each draw two socks out of the bag, without replacement. suppose all socks feel the same, so when two socks are drawn from the bag, all possibilities have equal probability. let m be the event that each person draws a matching pair of socks. what is the number of outcomes in m?

Step-by-step explanation:

There are 4 people and 4 different pairs of socks which we will denote by A,B,C and D. If every person ends with a matching pair of socks that means that:

Therefore the numbers of outcomes of M is

It is clear that each one of those outcomes have the same probability to happen. Then we will compute the probability of one of them (which we will denote as ) and then multiply by 24 to obtain the probability of M.

We will compute the probability of:

Without loss of generality we will assume that person 1 chooses first, then person 2 and so on.

Observe that the number of ways of choosing 2 socks is given by:

Therefore, the probability of person 1 to choose the pair A={1,2} is .

After that, there would remain 6 socks in the bag. Then, the probability of person 2 of choosing pair B={3,4} is 1 in . That is

After that, there would remain 4 socks in the bag. Then, the probability of person 3 of choosing pair C={5,6} is 1 in . That is

Finally when person 4 chooses, there would be only 2 socks (pair D={7,8}) so, the probability of choosing pair D is 1.

Therefore, the probability of to happen is

And in consequence

the translated point is (5,2)