Question

A bag contains $12$ pairs of socks. Four socks are picked at random. Find the probability that there is at least one pair.

Answer 1

In order to solve this question, we first of all find the ways in which four socks are drawn, then the number of ways in which a pair is drawn, and then by finding the probability that no pair is drawn, we can subtract it from $1$ which gives the probability that at least one pair is drawn.

Formula used: The formula used here:
To find $m$events from total number of events $n$
$\left( n-1 \right)\left( n-2 \right)....\left( n-m \right)$
To find the probability
The ratio is found between the ways of occurring of a particular event to the total number of ways.

Complete step-by-step solution:
$12$ pairs of socks means $24$socks are picked, so total number of ways in which the four socks can be picked up at random is
$24\times 23\times 22\times 21$
The total number of ways in which a pair is picked up is:
$24\times 22\times 20\times 18$
Therefore, the probability of not getting a single pair is the ratio of number of ways a pair is picked up to number of ways for picking four socks
$\dfrac{24\times 22\times 20\times 18}{24\times 23\times 22\times 21}=\dfrac{224}{323}$
Now, the probability that at least one pair is selected is found by subtracting probability for no pair from $1$

So, probability of getting at least one pair is $1-\dfrac{224}{323}=\dfrac{323-224}{323}=\dfrac{99}{323}$

Note: To find the probability of getting at least one pair of socks, we need to first find the probability of finding not getting a single pair of socks. This type of question can be implemented by taking the negation side of the problem and then subtracting the whole to the existing one.