2 balls are randomly put in 3 boxes, if 2 boxes are randomly selected, what is probability of getting 2 balls

To solve this problem, we need to consider the different ways the balls can be distributed in the boxes and then the probability of selecting the boxes with the balls.

Step 1: Distribution of balls in boxes There are 3 boxes and 2 balls. The balls can be distributed in the following ways:

• Both balls in one box: There are 3 ways for this to happen (one for each box).
• One ball in one box and the other ball in another box: There are 3 ways to choose the box for the first ball and 2 ways to choose the box for the second ball, so there are 3*2 = 6 ways for this to happen.

So, there are a total of 3 + 6 = 9 ways to distribute the balls in the boxes.

Step 2: Probability of selecting the boxes with the balls We are interested in the cases where we select the boxes with the balls. This can happen in the following ways:

• If both balls are in one box, we need to select this box and one of the other two boxes. There are 3 ways to choose the box with the balls and 2 ways to choose the other box, so there are 3*2 = 6 ways for this to happen.
• If the balls are in different boxes, we need to select these two boxes. There are 3 ways to choose the box for the first ball and 2 ways to choose the box for the second ball, so there are 3*2 = 6 ways for this to happen.

So, there are a total of 6 + 6 = 12 ways to select the boxes with the balls.

Step 3: Calculation of the probability The probability of selecting the boxes with the balls is the number of ways to select the boxes with the balls divided by the total number of ways to distribute the balls in the boxes. So, the probability is 12/9 = 4/3.

However, this result is greater than 1, which is not possible for a probability. This suggests that there is a mistake in the calculation. The mistake is in step 2: when the balls are in different boxes, we can't just multiply the number of ways to choose the box for the first ball by the number of ways to choose the box for the second ball, because the order in which we select the boxes doesn't matter. So, there are actually 3 ways to select the boxes with the balls when the balls are in different boxes.

So, the correct number of ways to select the boxes with the balls is 6 + 3 = 9, and the correct probability is 9/9 = 1. So, if 2 balls are randomly put in 3 boxes, the probability of getting 2 balls when 2 boxes are randomly selected is 1.