luka has a bag containing 5 socks: 3 red, 1 white, and 1 black. he draws 1 sock out of the bag, replaces it…

luka has a bag containing 5 socks: 3 red, 1 white, and 1 black. he draws 1 sock out of the bag, replaces it, and then draws another sock. what is the probability that he will draw a black sock and then a red sock, $p(black, then red)$?\n$\\frac{1}{25}$\n$\\frac{1}{15}$\n$\\frac{3}{15}$\n$\\frac{3}{25}$

luka has a bag containing 5 socks: 3 red, 1 white, and 1 black. he draws 1 sock out of the bag, replaces it, and then draws another sock. what is the probability that he will draw a black sock and then a red sock, $p(black, then red)$?\n$\\frac{1}{25}$\n$\\frac{1}{15}$\n$\\frac{3}{15}$\n$\\frac{3}{25}$

Answer

Explanation:

Step1: Calculate probability of drawing black sock

The probability of drawing a black sock on the first draw is the number of black socks divided by the total number of socks. There is 1 black sock and 5 total socks, so $P(\text{black})=\frac{1}{5}$.

Step2: Calculate probability of drawing red sock

Since the sock is replaced, the probability of drawing a red sock on the second draw is the number of red socks divided by the total number of socks. There are 3 red socks and 5 total socks, so $P(\text{red})=\frac{3}{5}$.

Step3: Calculate combined probability

For independent events (drawing with replacement makes the events independent), the probability of both events occurring is the product of their individual probabilities. So $P(\text{black, then red}) = P(\text{black})\times P(\text{red})=\frac{1}{5}\times\frac{3}{5}=\frac{3}{25}$.

Answer:

$\frac{3}{25}$