Question

select the correct answer.
jim’s 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 initial socks

Each pair has 2 socks:
Black: $2 \times 2 = 4$, White: $3 \times 2 = 6$, Green: $1 \times 2 = 2$, Gray: $2 \times 2 = 4$
Total initial socks: $4 + 6 + 2 + 4 = 16$

Step2: Socks left after first pick

He removed 1 black, 1 white, 1 gray.
Remaining: Black: $4-1=3$, White: $6-1=5$, Green: $2$, Gray: $4-1=3$
Total remaining socks: $3 + 5 + 2 + 3 = 13$

Step3: Favorable socks for a pair

To get a pair, he needs to pick black, white, or gray (matching the ones he already has).
Favorable count: $3 + 5 + 3 = 11$

Step4: Compute probability

Probability = $\frac{\text{Favorable socks}}{\text{Total remaining socks}}$
$\text{Probability} = \frac{11}{13}$

Answer:

B. $\frac{11}{13}$