Question

STAT Question :

1. We have 3 distinct balls (indexed Ball 1, Ball 2, Ball 3) and 3 distinct boxes (indexed Box 1, Box 2, and Box 3). Now randomly assign each ball (independently of other balls) into one of the boxes.
   Fully simplify your final answer to each question into a fraction or decimal.

1. Calculate the probability that no box is empty.
2. Calculate the probability that ONLY Box 1 is empty.

NEED HELP !! THANKS

Answer 1

Since balls and boxes all are distinct. So for each of 3 balls, there are 3 available choices.
So total no of ways, n(S) = 3•3•3 = 27

(i) If no box is empty, no of ways of arrangements of n distinct objects at n place = n!

So, favourable ways = 3! = 6

Hence, Pr(No box is empty) = 6/27 = 2/9

(ii) If box 1 has to be empty, for each ball, there are two choices, either box 2 or box 3. But all the balls cant go to a single box, so favourable ways = 2•2•2 - 2 = 8-2 = 6

Hence, Pr(Only box 1 is empty) = 6/27 = 2/9