We have been given that a bag contains 2 red marbles and 3 black marbles. Abby picks a marble without looking, returns it to the bag, and then draws a second marble. We are asked to find the probability that both marbles are red.
Let [tex]P(A)[/tex] be probability of getting a red marble on 1st draw and [tex]P(B)[/tex] be probability of getting a red marble on 2nd draw.
Number of red marbles = 2
Total number of marbles = [tex]2+3=5[/tex].
[tex]P(A)=\frac{\text{Number of red marbles}}{\text{total marbles}}[/tex]
[tex]P(A)=\frac{2}{5}[/tex]
Since Abby returns the marble into the bag, so number of marbles will not change. This means that probability of both events is independent.
[tex]P(B)=\frac{\text{Number of red marbles}}{\text{total marbles}}[/tex]
[tex]P(B)=\frac{2}{5}[/tex]
When two events are independent, then their probability is [tex]P(\text{A and B})=P(A)\times P(B)[/tex]
[tex]P(\text{A and B})=\frac{2}{5}\times \frac{2}{5}[/tex]
[tex]P(\text{A and B})=\frac{4}{25}[/tex]
Therefore, the probability that both marbles are red would be [tex]\frac{4}{25}[/tex].
