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

Hint: 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.