Expected number of cards given to people where the card matches the persons birthday.

There is a set of $$n$$ cards, numbered 1 to $$n$$. The cards are distributed at random to $$n$$ people. Assume that $$n$$ is large enough that we are sure that every person in the group has a card with his/her age on it.

(a) If there are $$n$$ cards and $$n$$ people, what is the expected number of cards that are given to people whose age matches the number on the card?

(b)Now, suppose there are $$n$$ cards and $$\frac{n}{k}$$ people, assuming that $$\frac{n}{k}$$ is an integer. Each of the $$\frac{n}{k}$$ people is given $$k$$ cards. What is the expected number of cards given to people such that the card matches the person's age?

(a) Call event $$A_i$$ the event that the $$i^{th}$$ person gets distributed a card with his/her age on it. Let $$X_i$$ be an indicator random variable that is equal to 1 if $$A_i$$ occurs and 0 if $$A_i$$ does not occur. Then, the number of cards that are distributed to people that match the age of the person, X, is given by:

$$X = \sum_{i = 1}^{n}{X_i}$$

So, the expected value of $$X$$ is the following:

$$E(X) = E\left(\sum_{i = 1}^{n}{X_i}\right) = \sum_{i = 1}^{n}{E(X_i)} = nP(A_i) = n \frac{1}{n} = 1$$

I say that the $$P(A_i)$$ is $$\frac{1}{n}$$ because if the cards are numbered $$1$$ to $$n$$, then for the $$i^{th}$$ person who only has $$1$$ age, there is only $$1$$ card out of the $$n$$ that has his/her age on it.

(b) So in the first case, I think my reasoning is valid because the number of people who have a card that matches their age is the same as the number of cards distributed to a person where the number on the card matches that person's age. I believe this is still valid in the second case because the cards are numbered $$1$$ to $$n$$ and each person has only one age, which is a number in the set $$1, ..., n$$.

Taking $$A_i$$, $$X_i$$, and $$X$$ to mean the same thing as part (a), the key question to answer is what is the $$P(A_i)$$, the probability that the $$i^{th}$$ person has a card with his/her age on it. The only difference being that the $$i^{th}$$ person has $$k$$ cards instead of $$1$$. I am kind of confused as to how to compute this probability.

There is still one card with the person’s age on it, and the person gets $$k$$ out of $$n$$ cards. At most one of them has their age on it, and the probability for one of them to be the one with that age is $$\frac kn$$; so the sum over the $$\frac nk$$ people is again $$1$$.