Question

select the correct answer.
jims sock drawer has two pairs of black socks, three pairs of white, one pair of green, and two pairs of gray. one evening he randomly picks three socks assuming that he will get a matched pair, but instead turns up with 1 black, 1 white, and 1 gray sock. without putting back the socks he picked, he picks another sock randomly. what is the probability that he will have a complete pair?
a. $\frac{2}{13}$
b. $\frac{11}{13}$
c. $\frac{4}{13}$
d. $\frac{1}{16}$

Explanation:

Step1: Calculate total number of socks

There are \(2\times2 + 3\times2+2\times1 + 2\times2=4 + 6+2 + 4 = 16\) socks. After picking 3 socks (1 black, 1 white, 1 gray), there are \(n = 16-3=13\) socks left.

Step2: Calculate probability of getting a complete pair

The number of non - matching socks to the three already picked is \(1\) (since there are pairs). The number of favorable cases (picking a sock that will make a pair) is \(4\) (because for each of the three non - matched socks, there is one sock left in the drawer to make a pair, and also, if we consider the pairs concept, after three non - paired socks are taken, there are \(4\) socks that can complete a pair). The probability \(P=\frac{4}{13}\)

Answer:

B. \(\frac{4}{13}\)