Question

Consider a collection of 10 empty boxes (numbered from 1 to 10) and 5 balls. Suppose that each ball is placed in a box chosen at random. Assume that this placement of a ball in a box is performed independently for each ball. Note that a box may contain more than one ball and that some of the boxes will necessarily remain empty. Hint: Since more than one ball can be placed in the same box, it is best to think of assigning a box to a ball. a. Find the probability that exactly two balls end up in box 1. b. Find the probability that each ball ends up in a different box (i.e. the probability that no two balls are placed in the same box).

Answer 1

there is a total of 10 boxes and 5 balls to be placed in them.

a box may contain more than one ball and that some of the boxes will necessarily remain empty ( since there are 5 balls and at least 5 boxes will remain empty).

1st ball can be placed in 10 boxes.

2nd ball also can be placed in 10 boxes.

3rd ball also can be placed in 10 boxes.

followed by the 4th and 5th ball

as more than 1 ball can be placed in a single box

Hence there is a total of ways to place 5 balls in 10 boxes.

a) the probability that exactly two balls end up in box 1.

the two balls in box 1 out of 5 balls can be chosen ii ways

the rest 3 balls can be placed in the rest 9 boxes in ways

Hence the total number of ways that exactly two balls end up in box 1 =

the probability that exactly two balls end up in box 1 =

b) the probability that each ball ends up in a different box:

The total number of ways that each ball ends up in a different box = 10 * 9 * 8 * 7 * 6 = 30240

the probability that each ball ends up in a different box=