To determine the probability that Jim will complete a pair after picking a fourth sock, let's go through the solution step by step:
1. Initial Pairs of Socks:
- Black socks: 2 pairs (4 socks)
- White socks: 3 pairs (6 socks)
- Green socks: 1 pair (2 socks)
- Gray socks: 2 pairs (4 socks)
Therefore, the total number of socks is:
[tex]\[
(2 + 3 + 1 + 2) \times 2 = 16 \text{ socks}
\][/tex]
2. Socks Picked Initially:
Jim picks 1 black, 1 white, and 1 gray sock.
3. Remaining Socks:
- Remaining black socks: [tex]\(4 - 1 = 3\)[/tex]
- Remaining white socks: [tex]\(6 - 1 = 5\)[/tex]
- Remaining green socks: [tex]\(2 \text{ (untouched)}\)[/tex]
- Remaining gray socks: [tex]\(4 - 1 = 3\)[/tex]
Now, the total remaining socks are:
[tex]\[
3 + 5 + 2 + 3 = 13 \text{ socks}
\][/tex]
4. Favorable Cases:
To complete a pair, he must pick another black, white, or gray sock. Therefore, the number of favorable cases (remaining socks of each color initially picked) is:
[tex]\[
3 \text{ (black)} + 5 \text{ (white)} + 3 \text{ (gray)} = 11 \text{ socks}
\][/tex]
5. Probability Calculation:
The probability of picking a sock that completes a pair is the number of favorable cases divided by the total number of remaining socks:
[tex]\[
\frac{11}{13}
\][/tex]
Therefore, the probability that Jim will have a complete pair after picking the fourth sock is [tex]\( \frac{11}{13} \)[/tex], which matches option B.
So, the correct answer is:
[tex]\[ \boxed{\frac{11}{13}} \][/tex]
