Probability and Expected Value of Placing Balls into Separate Bags

Question

There's this probability question that I'm having trouble wrapping my head around:

You have 4 balls each with a different colour (say red, green, blue, and yellow) and 4 bags with the corresponding colour (1 red, 1 green, 1 blue, and 1 yellow bag). Each bag can hold only one ball. The balls are now randomly placed into the bags. Then:

1) What is the probability that exactly 2 balls are placed into their respectively coloured bag; and

2) What is the expected number of balls that are placed into the respectively couloured bag?

Answer 1

This is a small enough problem that you can simply calculate everything by brute force.

• How many arrangements of the balls in the bags have all four balls correctly placed?
• How many arrangements have exactly $3$ balls correctly placed? (This one should be very easy.)
• How many arrangements have exactly $2$ balls correctly placed? (HINT: How many ways are there to choose which two balls are correctly placed? How many ways are there to place the other two balls?) From this you can immediately get the answer to the first question.
• How many arrangements have exactly one ball correctly placed? This one takes the most work. Once one ball is correctly placed, how many ways are there to place the other three so that none of them is in the right bag?

From these numbers you can calculate the probability that $k$ balls are correctly placed for $k=0,1,2,3,4$ and then use those to answer the second question.