Question

# Three balls are randomly dropped into three boxes. Assume that any ball is equally likely to...

Three balls are randomly dropped into three boxes. Assume that any ball is equally likely to fall into each box. Specify an appropriate sample space and determine the probability that exactly one box will be empty.

we have three balls adnd three boxes, so sample space = 3^3 = 27

Calculation for probability

let us assume that Box 3 is empty

then, probability that box 1 has 2 balls out of 3 is C(3,2) and box 2 has 1 ball only

total number of outcomes = C(3,2)*1 = {3!/[(3-2)!*2!]}*1 = 3

Similarly, when box 2 has 2 balls and box 1 has 1 ball, then possible outcomes = C(3,2)*1 = {3!/[(3-2)!*2!]}*1 = 3

total possible outcomes when box 3 is empty = 3 + 3 = 6 outcomes

so, the probability that exactly box 3 is empty = (favorable outcome)/(total outcome) = 6/27

but, we need to find the probability that exactly any one of the boxes will be empty

Events of getting box 1, box2 or box3 empty are mutually disjoin

so, required probability that exactly one box will be empty = 3*(6/27) = 6/9 or 2/3