The probability of selecting a blue marble and a red marble from a bag containing red, blue and green marbles is $\dfrac{1}{3}$ and $\dfrac{1}{5}$ respectively. If the bag contains 14 green marbles, then find :

(a) number of red marbles.

(b) total number of marbles in the bag.

As the bag contains red, blue and green marbles.

∴ Probability of selecting a red marble + Probability of selecting a blue marble + Probability of selecting a green marble = 1

⇒ $\dfrac{1}{3} + \dfrac{1}{5}$ + Probability of selecting a green marble = 1

⇒ $\dfrac{5 + 3}{15}$ + Probability of selecting a green marble = 1

⇒ Probability of selecting a green marble = $1 - \dfrac{8}{15}$

⇒ Probability of selecting a green marble = $\dfrac{15 - 8}{15}$

⇒ Probability of selecting a green marble = $\dfrac{7}{15}$.

$\therefore \dfrac{\text{No. of green marbles}}{\text{Total no. of marbles}} = \dfrac{7}{15} \\[1em] \Rightarrow \dfrac{14}{\text{Total no. of marbles}} = \dfrac{7}{15} \\[1em] \Rightarrow \text{Total no. of marbles} = \dfrac{15 \times 14}{7} = 30.$

(a) Given,

⇒ Probability of selecting a red marble = $\dfrac{1}{5}$.

$\therefore \dfrac{\text{No. of red marbles}}{\text{Total no. of marbles}} = \dfrac{1}{5} \\[1em] \Rightarrow \dfrac{\text{No. of red marbles}}{30} = \dfrac{1}{5} \\[1em] \Rightarrow \text{No. of red marbles} = \dfrac{30}{5} = 6.$

Hence, no. of red marbles = 6.

(b) Hence, total no. of marbles in the bag = 30.