What is the probability that each box will contain one ball if two indistinguishable balls are randomly put in one of two distinguishable boxes?

To me there seems to be 2 possible ways to reach an answer:

First Method: Each ball has a 1/2 chance of going into each box. So there is a 1/4 chance that both balls are assigned to Box #1, a 1/4 chance that both balls are assigned to Box #2, and a 2/4 or 1/2 chance that one ball is assigned to Box #1 and the other is assigned to Box #2. We arrive at an answer of 1/2 using this method.

Second Method: There are 3 total ways to distribute the balls to the boxes. For the boxes we can have
(2 balls , 0 balls), (0 balls , 2 balls) or (1 ball, 1 ball). Out of the 3 possibilities, 1 of them is favorable so using this method, we get an answer of 1/3.

Which approach is correct? Thanks!

 Jun 7, 2022

For the second method, you (correct me if I'm wrong) overlooked the fact that the probability of each of the possibilities may not be the same.

 Jun 7, 2022

Rainbow squirrel you are spot on.  Great answer.


The easiest way to look at this is that you pick up a ball and put it in any box.

probabiliy will be 1 since there will be a ball in a box and it does not matter which box.


How you have a second ball and a choice of 2 places on where to put it.

Prob that it goes into the empty box is 1/2


1  *  1/2   =  1/2

 Jun 7, 2022

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