To solve the problem, let's break it down step by step:
1. Initial Socks Count:
Jim's sock drawer contains the following pairs:
- 2 pairs of black socks: [tex]\(2 \times 2 = 4\)[/tex] black socks.
- 3 pairs of white socks: [tex]\(3 \times 2 = 6\)[/tex] white socks.
- 1 pair of green socks: [tex]\(1 \times 2 = 2\)[/tex] green socks.
- 2 pairs of gray socks: [tex]\(2 \times 2 = 4\)[/tex] gray socks.
Therefore, the initial total number of socks is:
[tex]\[
4 \text{(black)} + 6 \text{(white)} + 2 \text{(green)} + 4 \text{(gray)} = 16 \text{ socks}
\][/tex]
2. Socks Picked by Jim:
Jim picks 1 black, 1 white, and 1 gray sock. After picking these socks, the remaining socks are:
- Black socks: [tex]\(4 - 1 = 3\)[/tex]
- White socks: [tex]\(6 - 1 = 5\)[/tex]
- Green socks: [tex]\(2\)[/tex] (unchanged)
- Gray socks: [tex]\(4 - 1 = 3\)[/tex]
So the total number of socks left after Jim picks three socks is:
[tex]\[
3 \text{(black)} + 5 \text{(white)} + 2 \text{(green)} + 3 \text{(gray)} = 13 \text{ socks}
\][/tex]
3. Favorable Outcomes:
To complete a pair, Jim needs to pick either another black, white, or gray sock. The count of these socks is:
- Black socks: [tex]\(3\)[/tex]
- White socks: [tex]\(5\)[/tex]
- Gray socks: [tex]\(3\)[/tex]
Thus, the number of favorable outcomes is:
[tex]\[
3 \text{(black)} + 5 \text{(white)} + 3 \text{(gray)} = 11 \text{ favorable socks}
\][/tex]
4. Probability Calculation:
The probability that Jim will pick a sock that completes a pair is the number of favorable outcomes divided by the total number of remaining socks:
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of remaining socks}} = \frac{11}{13}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{11}{13}}
\][/tex]
