Question

A bag contains 8 red balls and some blue balls. If one ball is drawn randomly the probability of drawing a red ball to a blue ball are in the ratio 5 : 2, determine the probability of drawing a blue ball from the bag.

Answer 1

Let the number of blue balls be x.

Number of red balls = 8

∴ Total number of balls = (x + 8)

P(blue ball is drawn) = `x/(x + 8)`

P(red ball is drawn) = `8/(x + 8)`

According to the given condition, the probability of drawing a blue ball to a red ball are in the ratio 5 : 2.

∴ `("P"("blue ball is drawn"))/("P"("red ball is drawn")) = 5/2`

∴ `(x/(x  +  8))/(8/(x  +  8)) = 5/2`

∴ `x/(x + 8) = 5/2 xx 8/(x + 8)`

∴ x(x + 8) = 20(x + 8)

∴ x² + 8x = 20x + 160

∴ x² – 12x – 160 = 0

∴ x² – 20x + 8x – 160 = 0

∴ x(x – 20) + 8(x – 20) = 0

∴ (x – 20)(x + 8) = 0

∴ x – 20 = 0 or x + 8 = 0

∴ x = 20 or x = – 8

But, number of balls cannot be negative.

∴ x = 20

∴ P(blue ball is drawn) = `x/(x + 8)`

= `20/(20 + 8)`

= `20/28`

= `5/7`

∴ The probability that a blue ball is drawn from the bag is `5/7`.