2 balls are randomly put in 3 boxes, if 2 boxes are open, probability of getting 2 balls

To solve this problem, we need to understand the different scenarios that can occur when we randomly put 2 balls in 3 boxes.

Step 1: Calculate the total number of ways to distribute the balls We can put both balls in one box, or we can put each ball in a different box.

• If we put both balls in one box, there are 3 ways to do this (one for each box).
• If we put each ball in a different box, there are 3 ways to choose the box for the first ball and 2 ways to choose the box for the second ball, for a total of 3*2 = 6 ways.

So, the total number of ways to distribute the balls is 3 + 6 = 9.

Step 2: Calculate the number of ways to get 2 balls when opening 2 boxes We can get 2 balls when opening 2 boxes in two scenarios:

• Both balls are in one of the boxes we open. There are 3 ways to choose the box with both balls and 2 ways to choose the other box to open, for a total of 3*2 = 6 ways.
• Each box we open has one ball. There are 3 ways to choose the box for the first ball and 2 ways to choose the box for the second ball, for a total of 3*2 = 6 ways.

So, the total number of ways to get 2 balls when opening 2 boxes is 6 + 6 = 12.

Step 3: Calculate the probability The probability of getting 2 balls when opening 2 boxes is the number of ways to get 2 balls divided by the total number of ways to distribute the balls.

So, the probability is 12/9 = 1.33, which is greater than 1 and therefore not possible.

There seems to be a mistake in the problem setup or in the calculations. The number of ways to get 2 balls when opening 2 boxes should not be greater than the total number of ways to distribute the balls.