Expected Value using the indicator random variable

Question

Question: You invite 15 friends; thus, the total number of people at the party is equal to 16. You have bought an unlimited amount of cookies: 5 types C1;C2;C3;C4;C5 of chocolate and 3 types B1;B2;B3 of brownies. Each of the 16 students gets 3 cookies; each of these cookies is uniformly, and independently, chosen from the 8 types of cookies.

Define the random variable X;

X = the number of people who get exactly 2 chocolates

What is the expected value E(X) of the random variable X? [Use indicator random variables]

Answer: 7.03125

Attempt: I took for the indicator variable: X = 1 if people get exactly 2 chocolates and 0 for all other cases.

There has to be 8^3 total ways a person can get his cookie and 5^2 ways the person get the chocolate cookie. How do I incorporate the indicator variable to this?

Answer 1

Guide:

Let $X_i$ be $1$ if person $i$ get exactly $2$ chocolates and $0$ otherwise.

$$E[X_i]=Pr(X_i=1)=\binom32 p^2(1-p)$$

Can you determine the value of $p$?

Also, our quantity of interest is of the form of $E[\sum_{i=1}^n X_i]$.