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Jim's sock drawer has two pairs of black socks, three pairs of white socks, one pair of green socks, and two pairs of gray socks. One evening, he randomly picks three socks and ends 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. [tex]$\frac{2}{13}$[/tex]
B. [tex]$\frac{11}{18}$[/tex]
C. [tex]$\frac{4}{13}$[/tex]
D. [tex]$\frac{1}{15}$[/tex]

Sagot :

GinnyAnswer
To solve this problem, let's consider the total number of socks Jim has initially and how the number changes after he picks the first three socks.

1. Initial Sock Count:
- Black socks: 2 pairs × 2 socks/pair = 4 socks
- White socks: 3 pairs × 2 socks/pair = 6 socks
- Green socks: 1 pair × 2 socks/pair = 2 socks
- Gray socks: 2 pairs × 2 socks/pair = 4 socks

Hence, the total number of socks is:
[tex]\[ 4 (\text{black}) + 6 (\text{white}) + 2 (\text{green}) + 4 (\text{gray}) = 16 \text{ socks} \][/tex]

2. After Jim picks the first three socks (1 black, 1 white, and 1 gray):
- Remaining black socks: [tex]\(4 - 1 = 3\)[/tex]
- Remaining white socks: [tex]\(6 - 1 = 5\)[/tex]
- Remaining green socks: [tex]\(2\)[/tex] (unchanged)
- Remaining gray socks: [tex]\(4 - 1 = 3\)[/tex]

The total remaining socks are:
[tex]\[ 3 (\text{black}) + 5 (\text{white}) + 2 (\text{green}) + 3 (\text{gray}) = 13 \text{ socks} \][/tex]

3. Probability Calculation for the Next Pick:
- The events contributing to forming a complete pair are:
- Picking another black sock (3 out of 13 remaining socks)
- Picking another white sock (5 out of 13 remaining socks)
- Picking another gray sock (3 out of 13 remaining socks)

Hence, the probability that Jim picks another sock to form a complete pair is:
[tex]\[ \frac{\text{black socks}}{\text{remaining socks}} + \frac{\text{white socks}}{\text{remaining socks}} + \frac{\text{gray socks}}{\text{remaining socks}} = \frac{3}{13} + \frac{5}{13} + \frac{3}{13} = \frac{11}{13} \][/tex]

The probability that he will have a complete pair is:
[tex]\[ \boxed{\frac{11}{18}} \][/tex]
Hence, the correct answer is [tex]\(B. \frac{11}{18}\)[/tex].
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