Question

A closet contains 10 pairs of shoes. If 8 shoes are randomly selected, what is the probability that there will be (a) no complete pair? (b) exactly 1 complete pair?

Probability
ANSWERED
asked 2021-08-21
A closet contains 10 pairs of shoes.
If 8 shoes are randomly selected, what is the probability that there will be
(a) no complete pair?
(b) exactly 1 complete pair?

Expert Answers (1)

2021-08-22

(a)
Since there shouldn’t be a complete pair, we are allowed to choose at most 1 shoe from each pair.
Therefore, our choice can be made in the following way: first choose 8 pairs from the 10 pairs in the closet. this can be done in \(\displaystyle{\left(\begin{array}{c} {10}\\{8}\end{array}\right)}\) ways.
Then from chosen 8 pairs, choose one shoe from each.
This can be done in \(\displaystyle{2}^{{8}}\) ways as there are 2 choices for each pair.
Therefore, total number of choices is \(\displaystyle{\left(\begin{array}{c} {10}\\{8}\end{array}\right)}{2}^{{8}}.\).
The number of ways to randomly choose 8 shoes out of 20 is \(\displaystyle{\left(\begin{array}{c} {20}\\{8}\end{array}\right)}\). Therefore,
\(P(\text{no complete pair})=\frac{((10),(8))2^8}{((20),(8))}\)
(b)
For getting exactly one pair, we can first choose the pair which will appear completely. there are 10 ways of doin it.
Then we need to choose 6 shoes from 9 remaining pairs.
Proceeding as part(a) we see that there are ((9),(6))2^6 ways for this. Therefore, the total number is \(\displaystyle{\left(\begin{array}{c} {9}\\{6}\end{array}\right)}{2}^{{6}}\cdot{10}\). Thus,
\(P(\text{exactly one pair})=\frac{((9),(6))2^6*10}{((20),(8))}\)
Result: \(P(\text{no complete pair})=\frac{((10),(8))2^8}{((20),(8))}\)
\(P(\text{exactly one pair})=\frac{((9),(6))2^6*10}{((20),(8))}\)

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