To solve the problem of finding the probability of leaving the house with a white sock on your left foot and a brown sock on your right foot, we can break it down into two sequential probabilities:
1. Probability of picking a white sock first.
2. Probability of picking a brown sock second, given that a white sock was picked first.
Let's go through the detailed steps:
### Step 1: Probability of Picking a White Sock First
First, we determine the total number of socks and the number of white socks.
- Total socks: [tex]\(2 \, \text{(white)} + 4 \, \text{(brown)} + 2 \, \text{(black)} = 8 \)[/tex]
Since there are 2 white socks, the probability of picking a white sock first is:
[tex]\[ \text{Probability of white first} = \frac{2}{8} = \frac{1}{4} \][/tex]
### Step 2: Probability of Picking a Brown Sock Second (Given First Sock is White)
After picking one white sock, we are left with 7 socks:
- Remaining socks: [tex]\(1 \, \text{(white)} + 4 \, \text{(brown)} + 2 \, \text{(black)} = 7\)[/tex]
The number of brown socks remains 4. Therefore, the probability of picking a brown sock next is:
[tex]\[ \text{Probability of brown second} = \frac{4}{7} \][/tex]
### Step 3: Combined Probability of Both Events
To find the probability of both events happening (picking a white sock first and a brown sock second), we multiply the two probabilities together:
[tex]\[ \text{Combined probability} = \left(\frac{1}{4}\right) \times \left(\frac{4}{7}\right) = \frac{1 \times 4}{4 \times 7} = \frac{4}{28} = \frac{1}{7} \][/tex]
So, the probability that you leave the house with a white sock on your left foot and a brown sock on your right foot is [tex]\( \frac{1}{7} \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{7}} \][/tex]
