Expected Number Correct on Matching Test

Question

Many History and English tests will include a set of questions in which you have to match elements of one set to elements of another. For example, you may have a list of dates and a list of people, with the object of matching each person with the date he or she operated. My question is, if you don't study, so you fill the question out randomly, and there are n elements in each set, about how many will you get correct? What is the standard deviation? When I've brute-forced my way through the first few values, I've gotten an expected value of 1. Can someone prove/disprove that this stays the same.

Answer 1

Hint: The Linearity of Expectation works even if the random variables being summed are dependent. $$\mathsf E(\sum_{i=1}^n X_i)=\sum_{i=1}^n \mathsf E(X_i)$$ However, for the variance you do have to consider dependency. $$\mathsf E((\sum_{i=1}^n X_i)^2) = \sum_{i=1}^n \mathsf E(X_i^2)+\underset{i\neq j}{\sum_{i=1}^n\sum_{j=1}^n}\mathsf E(X_iX_j)$$

Now, when $X_i$ is the indicator random variable for the event of item $i$ being a match, $\mathsf E(X_i)=\mathsf P(X_i=1)$ and $\mathsf E(X_iX_j)=\mathsf P(X_i=1\cap X_j=1)$

Evaluat these probabilities, and solve for mean and variance of the sum (and hence the standard deviation).

Answer 2

Guide:

We usually use indicator variable for this type of question.

Let $X_i=1$ if the $i$-th element from the first set is matched correctedly and $0$ otherwise.

We want to compute $\mathbb{E}[\sum_{i=1}^nX_i]$

Try to use property of indicator variable and also linearity of expectation to solve the problem.

Answer 3

$$\mathsf E(\sum_{i=1}^n X_i)=\sum_{i=1}^n \mathsf E(X_i)$$

The chance of any one element being matched with the correct element from the other set is $1/n$, as there are $n$ possible answers, and only $1$ is correct. By the above equation, we get an answer of $1$.