Question

If 12 balls are thrown at random into 20 boxes, what is the probability that no box will receive more than one ball? Please explain.

Answer 1

Solution:

Probability of an event E is given by,

P(E) = number of favourable outcomes/total number of outcomes

Hence, probability that no ball will receive more than one ball = (number of ways of throwing 12 balls in 20 boxes such that no box receive more than one ball)/(total number of ways of throwing 12 balls into 20 boxes)

First we shall calculate the number of ways of throwing 12 balls in 20 boxes such that no box receive more than one ball.

For first ball we have total 20 boxes hence, first ball can be thrown by 20 ways.

Since, no box is required to receive more than one ball and one ball is already placed in any one box, therefore we have now 19 boxes in which 2nd ball can be thrown. Hence, 2nd ball can be thrown by 19 ways.

Since, no box is required to receive more than one ball and two balls are already placed in any two boxes, therefore we have now 18 boxes in which 3rd ball can be thrown. Hence, 3rd ball can be thrown by 18 ways.

Similarly, 4th ball can be thrown by 17 ways, 5th ball can be thrown by 16 ways, 6th ball can be thrown by 15 ways, 7th ball can be thrown by 14 ways, 8th ball can be thrown by 13 ways, 9th ball can be thrown by 12 ways, 10th ball can be thrown by 11 ways, 11th ball can be thrown by 10 ways and 12th ball can be thrown by 9 ways.

Hence, number of ways of throwing 12 balls in 20 boxes such that no box receive more than one ball is give by,

(20 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9)

Now we shall obtain total number of ways of throwing 12 balls into 20 boxes.

For each of the 12 balls, all the 20 boxes are available. Therefore, each of the 12 balls can be thrown by 20 ways.

Hence, total number of ways of throwing 12 balls into 20 boxes is given by,

Total = (20)¹²

Hence, probability that no balls will receive more than one ball is,

If 12 balls are thrown at random into 20 boxes, the probability that no box will receive more than one ball is 0.01473.