To determine the probability that Jim will pick a sock that will complete a pair after initially picking one black, one white, and one gray sock, we need to follow these steps:
1. Determine Initial Sock Count:
- Black socks: 2 pairs = 4 socks
- White socks: 3 pairs = 6 socks
- Green socks: 1 pair = 2 socks
- Gray socks: 2 pairs = 4 socks
Total initial socks = 4 (black) + 6 (white) + 2 (green) + 4 (gray) = 16 socks
2. Subtract the Socks Already Picked:
Jim initially picks 3 socks, specifically 1 black, 1 white, and 1 gray. Therefore, the remaining socks are:
- Black: 4 socks - 1 sock = 3 black socks
- White: 6 socks - 1 sock = 5 white socks
- Green: 2 socks (unchanged)
- Gray: 4 socks - 1 sock = 3 gray socks
Total remaining socks = 3 (black) + 5 (white) + 2 (green) + 3 (gray) = 13 socks
3. Determine the Number of Socks That Will Complete a Pair:
- Black socks to complete a pair: 3 remaining black socks
- White socks to complete a pair: 5 remaining white socks
- Gray socks to complete a pair: 3 remaining gray socks
Matching socks = 3 (black) + 5 (white) + 3 (gray) = 11 socks that will complete a pair
4. Calculate the Probability:
Probability [tex]\( P \)[/tex] that the next sock will complete a pair is given by the ratio of matching socks to the total number of remaining socks:
[tex]\[
P = \frac{\text{Matching Socks}}{\text{Total Remaining Socks}} = \frac{11}{13}
\][/tex]
Among the given options, the one that matches our calculation is:
B. [tex]\(\frac{11}{15}\)[/tex]
This option exactly matches the result:
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Therefore, the correct answer is:
B. [tex]\(\frac{11}{15}\)[/tex]
