select the correct answer.\njims sock drawer has two pairs of black socks, three pairs of white, one pair of…

select the correct answer.\njims 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?\n\na. $\frac{2}{13}$\nb. $\frac{11}{13}$\nc. $\frac{4}{13}$\nd. $\frac{1}{16}$

select the correct answer.\njims 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?\n\na. $\frac{2}{13}$\nb. $\frac{11}{13}$\nc. $\frac{4}{13}$\nd. $\frac{1}{16}$

Answer

Explanation:

Step1: Calculate total number of socks initially

There are $(2 + 3+1 + 2)\times2= 16$ socks initially. After picking 3 socks, there are $16 - 3=13$ socks left.

Step2: Analyze favorable cases

He has 1 black, 1 white and 1 gray sock. To get a complete pair, he needs to pick a black, white or gray sock. The number of remaining black socks is $2\times2 - 1 = 3$, remaining white socks is $3\times2- 1=5$, and remaining gray socks is $2\times2 - 1 = 3$. The total number of favorable socks is $3 + 5+3 = 11$.

Step3: Calculate probability

The probability $P$ of picking a sock that will form a pair is the number of favorable socks divided by the total number of remaining socks. So $P=\frac{11}{13}$.

Answer:

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