Question

In your drawer of socks, there are 8 red, 6 blue, 4 white and 2 black socks. This morning, you grabbed one sock but put it back because you didn't want to wear that color today. Then, you grabbed a second sock and then a third sock. (Assume you put each one back before grabbing the next one.)
What is the chance you grabbed three red socks?

Answer 1

To determine the probability of grabbing three red socks in this scenario, we first need to know the total number of socks available, as well as how many of those socks are red.

You have the following socks:
- 8 red
- 6 blue
- 4 white
- 2 black

The total number of socks is:
\[
8 + 6 + 4 + 2 = 20 \text{ socks}
\]

The probability of grabbing a red sock on a single attempt is given by the ratio of the number of red socks to the total number of socks:
\[
P(\text{red sock}) = \frac{8}{20} = \frac{2}{5}
\]

Now we want to find the probability of grabbing three red socks in a row. Since you are putting back each sock after you grab it, each grab is independent, and the probability remains the same for each grab.

Thus, the probability of grabbing three red socks in succession is:
\[
P(\text{3 red socks}) = P(\text{red sock}) \times P(\text{red sock}) \times P(\text{red sock}) = \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right)
\]

Calculating this gives:
\[
P(\text{3 red socks}) = \left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125}
\]

Therefore, the probability of grabbing three red socks is:
\[
\boxed{\frac{8}{125}}
\]