Question

A player puts 3 balls into 4 empty baskets, what's the probability that the maximum number of balls in a single basket is exactly 2?

Answer 1

SOLUTION:-

Here number of ways of putting 3 balls into 4 empty baskets is calculated as follows:

For each of the 3 balls there are 4 options of baskets to put the ball. i.e, first ball can be put in 4 ways, second ball can be put in 4 ways and third ball also can be put in 4 ways. Thus there are total 4x4x4=$\mathbf{4^3}$ ways.

Now to find the number of favourable cases. i.e, here to find the number of ways of getting number of balls in a single basket is exactly 2.

This can happen if the first ball and second ball in one basket and the third in any of the other baskets or first, third ball are put in the same basket or second is put in any of the other baskets or second and third are put in one of the same basket and first is on any of the other baskets.

first ball can be put in 4 ways, then second when to put in the same basket there is only one way and third can be put in other 3 baskets in 3 ways. total 4x1x3=12 ways.

Similarly for the other two cases: we have 4x3x1 and 4x1x3 cases

so total ways is, 3x12=36 ways.

Hence the required probability = No. of favourable cases/Total cases

   $\mathbf{=\frac{36}{4^3}=\frac{9}{16}}$