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# Probability with balls and boxes

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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

#1
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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
#2
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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