Question

olivia has a bag of marbles. the bag contains 2 blue marbles, 1 red marble, and 2 green marbles. without looking, olivia picks a marble, returns the marble to the bag, and picks another marble. what is the probability, represented as a fraction, that olivia picks a blue marble both times?

Explanation:

Step1: Find total number of marbles

Total marbles = 2 (blue) + 1 (red) + 2 (green) = 5.

Step2: Probability of picking blue first

Probability of blue on first pick = $\frac{\text{Number of blue marbles}}{\text{Total marbles}}$ = $\frac{2}{5}$.

Step3: Probability of picking blue second (with replacement)

Since we replace the marble, the probability of blue on the second pick is also $\frac{2}{5}$.

Step4: Multiply the two probabilities

For independent events, the probability of both events happening is the product of their individual probabilities. So, probability of blue both times = $\frac{2}{5} \times \frac{2}{5}$ = $\frac{4}{25}$.

Answer:

$\frac{4}{25}$