Sure, let's solve the problem step by step.
You have the following socks in the drawer:
- 2 white socks
- 4 brown socks
- 2 black socks
First, we'll calculate the total number of socks:
[tex]\[ \text{Total number of socks} = 2 \text{ (white)} + 4 \text{ (brown)} + 2 \text{ (black)} = 8 \][/tex]
Next, we'll determine the probability of picking a white sock first:
[tex]\[ P(\text{white sock first}) = \frac{2}{8} = \frac{1}{4} \][/tex]
Now, with one sock already picked (one white sock), there will be 7 socks left in the drawer.
[tex]\[ \text{Remaining socks} = 1 \text{ (white)} + 4 \text{ (brown)} + 2 \text{ (black)} = 7 \][/tex]
Then we calculate the probability of picking a brown sock second:
[tex]\[ P(\text{brown sock second}) = \frac{4}{7} \][/tex]
To find the probability of both events occurring (picking a white sock first and a brown sock second), we multiply the probabilities of the two independent events:
[tex]\[ P(\text{white sock first and brown sock second}) = \left(\frac{1}{4}\right) \times \left(\frac{4}{7}\right) = \frac{1}{4} \times \frac{4}{7} = \frac{4}{28} = \frac{1}{7} \][/tex]
Thus, the probability that you will 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]
So, the correct answer is:
[tex]\[ \boxed{\frac{1}{7}} \][/tex]
