Statistics - expected number Suppose the probability of success obtaining heads is $0.5$, number of trials $n=10$. In a class of $100$ students, each student tosses a coin (assumed fair) $10$ times. Write down the expected number of students who will obtain $0,1,\ldots , 10$ heads.
If I start with $0$ heads, I understand that I need to calculate the probability of $0$ students obtaining $0$ heads, then the probability of $1$ student obtaining $0$ heads and so on until $100$. Then multiply the probability by the number of students and add all $100$ cases to get the expected number. Now obviously this is a lot of work, and this is just a part of a question, so is there something I don't understand or know, or is it the question?

Fortunately you don't need to do all that work. You can put to use the linearity of expectation. Denote the probability that a single student obtains $k$ heads by $p_k$. Then the expected number of students that this student contributes to the total number of students who obtain $k$ heads is $p_k\cdot1+(1-p_k)\cdot0=p_k$. Since the total number of students who obtain $k$ heads is the sum of $100$ contributions from the $100$ students, with each contribution given by $p_k$, by linearity of expectation the expected number of students who obtain $k$ heads is $100p_k$.
