Question

A drawer contains 8 black socks and 8 white. Two are pulled out randomly:
i. The probability that it is the same color?
ii. How many will have to be pulled out for matching pairs?

Answer 1

Hint: To solve this question first we find the probability of getting socks of any color but the next socks must be of the same color so then we find the probability of getting the same socks after removing the first socks and then multiply both the probability to get the final answer.

Complete step-by-step answer:
We were given 8 black socks and 8 white socks.
To find the probability of getting two same color socks if we pull one by one and the number of pulled for matching pairs.
Let, $$p_b$$ indicates the probability of getting a black ball. And $$p_w$$ indicates the probability of getting a white ball.

i. We have to find the probability of getting the same color socks.
Probability of pulling out one black socks is the ratio of the number of black socks to the total number of socks.
$$p_b={\displaystyle \frac{numberofblacksocks}{totalnumberofsocks}}$$
$$p_{b1}={\displaystyle \frac{8}{16}}$$
Now one black sock is removed so the total number of black socks remaining is only 7 and the total number of socks is 15.
Probability of again getting one black socks.
$$p_{b2}={\displaystyle \frac{7}{15}}$$
Total number of getting same color socks are $$p_{b1}\times p_{b2}$$
Probability of getting black color socks is $$p_b={\displaystyle \frac{7}{30}}$$
Similarly probability of getting white color socks is $$p_w={\displaystyle \frac{7}{30}}$$
hence the probability that it is the same color is
$P=p_b+p_w$
$\Rightarrow P={\displaystyle \frac{7}{30}}+{\displaystyle \frac{7}{30}}$
$\Rightarrow P={\displaystyle \frac{7}{15}}$

ii. We must pull out three socks so that they are matching pairs. Observe that in such a case even if the first two socks are of different colors, the third must be of either color and we have got a matching pair.

Note: To solve these types of questions students must know the concept of probability and conditional probability also. Students often make mistakes in conditional probability. In conditional probability, we have to consider the first case and remove the first object from total outcomes.