Question

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, find the number of blue balls in the bag?
[a] 5
[b] 8
[c] 10
[d] 6

Answer 1

Hint: Probability of event E = $\dfrac{n(E)}{n(S)}=\dfrac{\text{Favourable cases}}{\text{Total number of cases}}$ where S is called the sample space of the random experiment. Assume that the number of blue balls in the bag be x. Assume E be the event that the ball drawn is red and F the event that the ball drawn is blue. Find P(E) and P(F). Use the fact that the probability of drawing a blue ball is double that of a red ball. Hence form an equation in x. Solve for x. The value of x gives the number of blue balls in the bag.

Complete step-by-step answer:
Let the number of blue balls in the bag be x
Let E be the event: The ball drawn is red
Let F be the event: The ball drawn is blue
Since there are 5 red ball, the number of favourable cases to E is 5
Hence, we have n (E) = 5
The total number of ways in which we can choose a ball from x+5 balls is equal to x+5
Hence, we have n (S) = x+5
Since there are x blue balls, the number of favourable cases to F is x.
Hence, P (E) = $\dfrac{5}{5+x}$ and $P\left( F \right)=\dfrac{x}{x+5}$
Since the probability of drawing a blue ball is double that of drawing a red ball, we have
$\begin{aligned} & P\left( F \right)=2P\left( E \right) \\\ & \Rightarrow \dfrac{x}{x+5}=\dfrac{10}{x+5} \\\ \end{aligned}$
Multiplying both sides by x+5, we get
x = 10
Hence the number of blue balls in the bag is 10
Hence option [c] is correct.

Note: [1] It is important to note that drawing uniformly at random is important for the application of the above problem. If the draw is not random, then there is a bias factor in drawing, and the above formula is not applicable. In those cases, we use the conditional probability of an event.
[2] Verification:
Probability of drawing a blue ball $=\dfrac{10}{15}=\dfrac{2}{3}$
Probability of drawing a red ball $=\dfrac{5}{15}=\dfrac{1}{3}$
Clearly, the probability of drawing a blue ball is twice the probability of drawing a red ball.
Hence our answer is verified to be correct.