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### Algebra 2-2 ECAPost Test

#### Select the correct answer:

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, assuming that he will get a matched pair, but instead 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}{18}[/tex]
B. [tex]\frac{11}{15}[/tex]
C. [tex]\frac{4}{18}[/tex]
D. [tex]\frac{1}{16}[/tex]



Answer :

GinnyAnswer
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:
```
```
Therefore, the correct answer is:
B. [tex]\(\frac{11}{15}\)[/tex]

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