Question

There are five marbles in a bag. Three are green and two are yellow. You draw a marble, replace it, and then draw another. What is the probability of choosing two yellow marbles?

Answer 1

To determine the probability of drawing two yellow marbles in succession with replacement, we start by looking at the number of marbles of each color:

- Total marbles = 5 (3 green + 2 yellow)

Next, we calculate the probability of drawing a yellow marble in a single draw:

\[
P(\text{Yellow}) = \frac{\text{Number of yellow marbles}}{\text{Total number of marbles}} = \frac{2}{5}
\]

Since the draws are independent (we replace the marble after each draw), the probability of drawing a yellow marble both times can be calculated by multiplying the probability of drawing a yellow marble in the first draw by the probability of drawing a yellow marble in the second draw:

\[
P(\text{Yellow on 1st draw}) = \frac{2}{5}
\]

\[
P(\text{Yellow on 2nd draw}) = \frac{2}{5}
\]

Therefore, the combined probability of drawing two yellow marbles is:

\[
P(\text{Two Yellow Marbles}) = P(\text{Yellow on 1st draw}) \times P(\text{Yellow on 2nd draw}) = \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}
\]

Thus, the probability of choosing two yellow marbles is:

\[
\boxed{\frac{4}{25}}
\]