Question

a bag contains 9 green marbles, 3 red marbles, and 8 blue marbles. one marble is taken from the bag and put back after checking its color. a second marble is then taken out. what is the probability that the first is blue and the second green? a \\( \frac { 9 } { 50 } \\) b \\( \frac { 17 } { 20 } \\) c \\( \frac { 1 } { 19 } \\) d \\( \frac { 18 } { 95 } \\)

Explanation:

Step1: Calculate the probability of drawing a blue marble first

The total number of marbles is \(9 + 3+8=20\).
The probability of drawing a blue marble \(P(B)=\frac{8}{20}\) (number of blue marbles over total marbles).

Step2: Calculate the probability of drawing a green marble second

Since the marble is replaced, the total number of marbles is still 20.
The probability of drawing a green marble \(P(G)=\frac{9}{20}\) (number of green marbles over total marbles).

Step3: Use the multiplication rule for independent events

Since the two draws are independent (because the marble is replaced), the probability that the first is blue and the second is green is \(P = P(B)\times P(G)\).
Substitute the values: \(P=\frac{8}{20}\times\frac{9}{20}=\frac{72}{400}=\frac{9}{50}\)

Answer:

A. \(\frac{9}{50}\)