Question

a bag contains 15 marbles. the probability of randomly selecting a green marble is $\frac{1}{5}$. the probability of randomly selecting a green marble, replacing it, and then randomly selecting a blue marble is $\frac{2}{25}$. how many blue marbles are in the bag?
2
3
5
6

Explanation:

Step1: Find the number of green marbles

Let the number of green marbles be $g$. The probability of selecting a green marble $P(G)=\frac{g}{15}$, and $P(G)=\frac{1}{5}$. So, $\frac{g}{15}=\frac{1}{5}$, then $g = 15\times\frac{1}{5}=3$.

Step2: Use the compound - probability formula

The probability of two independent events (selecting a green marble and then a blue marble with replacement) is $P(G)\times P(B)$. We know $P(G)\times P(B)=\frac{2}{25}$ and $P(G)=\frac{1}{5}$. Let the number of blue marbles be $b$. Then $P(B)=\frac{b}{15}$. Substitute $P(G)$ into $P(G)\times P(B)=\frac{2}{25}$, we get $\frac{1}{5}\times\frac{b}{15}=\frac{2}{25}$.

Step3: Solve for the number of blue marbles

Cross - multiply the equation $\frac{1}{5}\times\frac{b}{15}=\frac{2}{25}$: $\frac{b}{75}=\frac{2}{25}$. Cross - multiply again: $25b=2\times75$, so $25b = 150$, and $b = 6$.

Answer:

6