Question

(Peer review) 30 students arrive in the exercise, each returning their homework answers to the assistant. Assistant mixes the response papers carefully and
Then share them back to students for review, one for each. Determine the expectation value for the number of students who decide to check their own response paper.
Tip: Define an indicator variable

X_i =( 1 if i student checks their own paper,
0, otherwise)
The linearity of the expected value may also be helpful.)

Answer 1

Let X be the number of students who decide to check their own response paper.

To find E(X).

By the problem, .

Hence by linearity of expectation, , since X_i's are indicators, for all i.

Now, any of the 30 papers can go to any of 30 students, and hence total number of possibilities = 30!.

For any fixed i, the i^th student will receive his (or her) paper if the remaining are distributed randomly among the rest of the students, which can be done in 29! ways.

Hence,

Thus,