Question

A man has 3 pairs of black socks and 2 pairs of brown socks kept together in a box. If he dressed hurriedly in the dark, the probability that after he has put on a black sock, he will then put on another black sock is
(a) $\dfrac{1}{3}$
(b) $\dfrac{2}{3}$
(c) $\dfrac{3}{5}$
(d) $\dfrac{2}{15}$

Answer 1

Hint: A pair is a set of two things they find the number of socks in the box. The probability of wearing a black sock ${{1}^{st}}$ is the number of black socks by total number of socks. Now the number of black socks reduces by 1. Now find probability of wearing black socks ${{2}^{nd}}$ time and take the product of ${{1}^{st}}$ and ${{2}^{nd}}$ to obtain total probability.

Complete step-by-step answer:
It is said that a man has 3 pairs of black socks. We know that 1 pair contains two socks. Thus the total number of black socks will be 6 as there are 3 pairs.
Similarly it is told that the man has 2 pairs of brown socks. Thus there are a total of 4 brown socks. And both these colors of socks are kept together in a box.
Thus, the total number of socks in the box = Number of black socks + Number of brown socks
The total number of socks in the box = 6 + 4 = 10
$\therefore$ Total number of socks in the box = 10
First let us find the probability of wearing a black sock first = Number of black socks / Total number of socks = $\dfrac{6}{10}=\dfrac{3}{5}$
Now let us find the probability of wearing black socks second time. The man had already worn one sock. Thus the number of socks in the box will be 9. He has worn a sock of black color. Thus the number of black socks in the box will be 5.
$\therefore$ Probability of wearing a black color socks = number of socks at present / total number of socks.
$\therefore$ Probability of wearing ${{2}^{nd}}$ black sock = $\dfrac{5}{9}$.
$\therefore$ Probability of wearing first black and again a black sock =
Probability of wearing ${{1}^{st}}$ black sock $\times$ Probability of wearing ${{2}^{nd}}$ black sock
$=\dfrac{3}{5}\times \dfrac{5}{9}=\dfrac{3}{9}=\dfrac{1}{3}$.
Thus we got the probability as $\dfrac{1}{3}$.
$\therefore$ Option (a) is the correct answer.

Note: After finding the probability of wearing ${{1}^{st}}$ black sock, remember to reduce 1 sock from the number of black socks and total number of socks. It is important that you find the total number of socks from the 5 pairs given. The probability you find in pairs will be wrong.