To determine the probability that Richard will select a black sock first and then a brown sock, we need to follow these steps:
1. Count the total number of socks:
- Black socks: 4
- Brown socks: 2
- Blue socks: 2
- Total socks = 4 + 2 + 2 = 8
2. Find the probability of selecting a black sock first:
- There are 4 black socks out of the total 8 socks.
- Probability (Black first) = [tex]\( \frac{4}{8} = 0.5 \)[/tex]
3. Find the probability of selecting a brown sock second, given a black sock was selected first:
- After selecting a black sock, there are 7 socks left (since one black sock has been removed).
- There are still 2 brown socks left.
- Probability (Brown second given Black first) = [tex]\( \frac{2}{7} \approx 0.2857 \)[/tex]
4. Calculate the overall probability of selecting a black sock first and then a brown sock:
- The combined probability is found by multiplying the individual probabilities:
[tex]\[
\text{Total probability} = \left( \frac{4}{8} \right) \times \left( \frac{2}{7} \right) = 0.5 \times 0.2857 = 0.14285714285714285
\][/tex]
So, the probability that Richard will select a black sock and then a brown sock is approximately [tex]\(0.14285714285714285\)[/tex]. To find this probability in fraction form, we note that it is equivalent to:
[tex]\[
\frac{1}{7}
\][/tex]
Thus, the correct answer is [tex]\(\frac{1}{7}\)[/tex].
