Question

a drawer contanes two pairs of red sockes, three pairs of yellow sockes, and one pair of black sockes. one sock is taken from the drawer and put back after checking its color. a second sock is taken out. what is the probability that the first sock and the second sock are yellow (answer is fractions)

Answer 1

To find the probability that both socks drawn are yellow, we first need to determine the total number of socks and the number of yellow socks.

1. **Total number of socks**:
- Red socks: 2 pairs \( \Rightarrow 2 \times 2 = 4 \) socks
- Yellow socks: 3 pairs \( \Rightarrow 3 \times 2 = 6 \) socks
- Black socks: 1 pair \( \Rightarrow 1 \times 2 = 2 \) socks

Total socks = \( 4 + 6 + 2 = 12 \) socks

2. **Number of yellow socks**: There are 6 yellow socks.

3. **Calculating the probability**:
- The probability that the first sock drawn is yellow is:
\[
P(\text{First sock is yellow}) = \frac{6}{12} = \frac{1}{2}
\]

- Since the sock is put back in the drawer, the total number of socks remains the same for the second draw. Therefore, the probability that the second sock drawn is yellow is also:
\[
P(\text{Second sock is yellow}) = \frac{6}{12} = \frac{1}{2}
\]

4. **Finding the joint probability** of both events (that both the first and second socks are yellow):
Using the multiplication rule for independent events:
\[
P(\text{First sock is yellow and Second sock is yellow}) = P(\text{First sock is yellow}) \times P(\text{Second sock is yellow}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]

Therefore, the probability that both the first sock and the second sock are yellow is:
\[
\boxed{\frac{1}{4}}
\]