To determine the probability that Luka will draw a black sock and then a red sock, we need to consider the fact that he is replacing the sock after each draw. This means each draw is an independent event, and the probabilities do not change between draws.
1. Calculate the probability of drawing a black sock first:
- There is a total of 5 socks.
- There is 1 black sock.
- The probability, [tex]\( P_{\text{black}} \)[/tex], is:
[tex]\[
P_{\text{black}} = \frac{\text{number of black socks}}{\text{total number of socks}} = \frac{1}{5} = 0.2
\][/tex]
2. Calculate the probability of drawing a red sock second:
- There are still 5 socks (since he replaces the first sock before drawing again).
- There are 3 red socks.
- The probability, [tex]\( P_{\text{red}} \)[/tex], is:
[tex]\[
P_{\text{red}} = \frac{\text{number of red socks}}{\text{total number of socks}} = \frac{3}{5} = 0.6
\][/tex]
3. Calculate the combined probability of both events happening in order:
- Since the draws are independent events, the joint probability is the product of the individual probabilities:
[tex]\[
P_{\text{black then red}} = P_{\text{black}} \times P_{\text{red}} = \left(\frac{1}{5}\right) \times \left(\frac{3}{5}\right) = \frac{3}{25}
\][/tex]
So, the probability that Luka will draw a black sock and then a red sock is [tex]\( \frac{3}{25} \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{3}{25}}
\][/tex]
