Question

Jack has a collection of 10 pairs of gloves in his wardrobe. Before a business trip, he has to pack his luggage, and he selects 8 gloves, without looking at them. We assume that any set of 8 gloves is equally likely to be chosen. Find the probability that these 8 gloves do not include any matching pair of gloves, that is, that there are no two (left and right) gloves, coming from the same pair.

Answer 1

Answer:

$\frac{6\cdot 8 \cdot 10 \cdot 12}{13 \cdot 15 \cdot 17\cdot 19}\approx 0.091$

Step-by-step explanation:

To compute the probability of not including any matching pairs, we can compute the number of ways in which he could pick 8 gloves with no matching pairs, and divide it by the total number of ways in which he could pick 8 gloves.

The process of choosing 8 gloves with no matching pair can be seen as follows:

He first picks any random gloves out of the 20, in this 1st step he has then 20 available choices. Then he needs to pick some other glove, BUT it cannot be the other glove from the pair he already picked one from. So at this step there aren't 19 choices, but 18 available choices. Now he has 2 gloves from different pairs, and needs to pick another glove. There are 18 gloves left, BUT he cannot pick the remaining glove from any of the 2 pairs he already has chosen one from. Therefore he only has 16 choices left. The process continues like this, until he chooses 8 gloves in total. Hence the total number of ways to choose 8 gloves with no matching pair is

$20 \cdot 18 \cdot 16 \cdot 14 \cdot 12 \cdot 10 \cdot 8 \cdot 6$

Now, the total numer of ways in which he could pick any 8 gloves out of 20 can be seen as follows: At the start he has 20 available gloves, he chooses any of them, so having 20 available choices on that first step. He then needs to choose any other glove, so he has 19 choices. He then picks any other glove, so he has 18 choices. And so on, until he has chosen the 8 gloves. Hence the total number of ways to choose 8 gloves is

$20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13$

Therefore the probability of choosing 8 gloves with no matching pair is

$\frac{20 \cdot 18 \cdot 16 \cdot 14 \cdot 12 \cdot10 \cdot 8 \cdot 6}{20 \cdot 19 \cdot 18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13}=\frac{6\cdot 8 \cdot 10 \cdot 12}{13 \cdot 15 \cdot 17\cdot 19}\approx 0.091$