Question

A bag contains 6 red balls and some blue balls, if the probability of drawing a blue ball from the bag is twice that of a red ball, find the number of balls in the bag.

Answer 1

Hint: Assume the number of blue balls be x. Now, the total number of balls that we have is (x+6) balls. Probability of drawing red balls is \[\dfrac{6}{x+6}\] . We have the probability of drawing blue balls equal to \[\dfrac{x}{x+6}\] . Now, according to the question, it is given that the probability of drawing a blue ball from the bag is twice that of a red ball. Now, solve the equation further.

Complete step-by-step answer:
According to the question, it is given that,
Number of red balls = 6 .
Let us assume the number of blue balls is x.
Total number of balls in the bag= x+6 ………………(1)
We have the number of red balls equal to 6 and number of total balls equal to (x+6) .
Probability of drawing red balls from the bag = \[\dfrac{number\,of\,red\,balls}{total\,number\,of\,balls}\] .
Probability of drawing red balls from the bag = \[\dfrac{6}{x+6}\] ……………..(2)
Probability of drawing blue balls from the bag = \[\dfrac{number\,of\,blue\,balls}{total\,number\,of\,balls}\] .
Probability of drawing blue balls from the bag = \[\dfrac{x}{x+6}\] …………………..(3)
According to the question, it is given that the probability of drawing a blue ball from the bag is twice that of a red ball.
Probability of drawing blue balls from the bag = 2 $\times$ Probability of drawing red balls from the bag
\[\Rightarrow \dfrac{x}{x+6}=2.\dfrac{6}{x+6}\]
Cancelling the term (x+6) on both sides and solving, we get
\[\Rightarrow x=12\]
So, the number of blue balls is 12.
Total number of balls in the bag = 12+6=18.
Hence, the total number of balls in the bag is 18.

Note: In this question one may write,
2 $\times$ Probability of drawing blue balls from the bag = Probability of drawing red balls from the bag. This expression is completely wrong. Because the probability of drawing blue balls is already twice that of a red ball. So, to make the probabilities equal we have to multiply by 2 in the probability of drawing red balls from the bag.