Question

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(\text{black, then red}) )? ( \frac{1}{25} ) ( \frac{1}{15} ) ( \frac{3}{15} ) ( \frac{3}{25} )

Explanation:

Step1: Calculate the probability of drawing a black sock

The total number of socks is \(n = 5\), and the number of black socks is \(m_1=1\). The probability of drawing a black sock \(P(\text{black})=\frac{m_1}{n}=\frac{1}{5}\)

Step2: Calculate the probability of drawing a red sock

Since the sock is replaced, the total number of socks is still \(n = 5\), and the number of red socks is \(m_2 = 3\). The probability of drawing a red sock \(P(\text{red})=\frac{m_2}{n}=\frac{3}{5}\)

Step3: Calculate the probability of the combined event

For independent events (because of replacement), \(P(A\text{ then }B)=P(A)\times P(B)\). Here \(A\) is drawing a black sock and \(B\) is drawing a red sock. So \(P(\text{black, then red})=\frac{1}{5}\times\frac{3}{5}=\frac{3}{25}\)

Answer:

\(\frac{3}{25}\) (the fourth option)