Question

A jar contains 54 marbles, each of which some are blue, some are green and some are white. The probability of selecting a blue marble at random is $\frac{1}{3}$ and the probability of selecting a green marble at random is $\frac{4}{9}$. How many white marbles does the jar contain?

Answer 1

Total number of marbles = 54.

It is given that, P(getting a blue marble) = $\frac{1}{3}$ and P(getting a green marble) = $\frac{4}{9}$

Let P(getting a white marble) be x.

Since, there are only 3 types of marbles in the jar, the sum of probabilities of all three marbles must be 1.

$$\begin{gathered} \therefore \frac{1}{3}+\frac{4}{9}+x=1 \\ \Rightarrow \frac{3+4}{9}+x=1 \\ \Rightarrow x=1-\frac{7}{9} \\ \Rightarrow x=\frac{2}{9} \end{gathered}$$
∴ P(getting a white marble) = $\frac{2}{9}$ ...(1)

Let the number of white marbles be n.

Then, P(getting a white marble) = $\frac{n}{54}$ ... (2)

From (1) and (2),
$$\begin{gathered} \frac{n}{54}=\frac{2}{9} \\ \Rightarrow n=\frac{2\times 54}{9} \\ \Rightarrow n=12 \end{gathered}$$

Thus, there are 12 white marbles in the jar.