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].
