# Expectation value of a balls problem

Suppose person A chooses randomly 3 balls from a basket which has 10 balls (when the balls are numbered! which means different from each other and uniformly distributed). Afterwards person A returns the balls and person B chooses 3 balls randomly and independently from the same basket.

a) What is is the expectation value of the balls which were chosen by both of them (chosen twice).

b) What is the expectation value of the number of balls chosen once from one of the persons?

I assumed $p(1)=p(2)=...=p(10)= \frac{1}{10}$. How can the answer be independent from parameters, when the expectation value depends on random variable? Cause let's assume person A picked 3 balls (probability of 3/10), shouldn't the answer include the random variable in order to compare the two?what am is missing?am i supposed to count every possible way it happens ?

a) Any given ball is chosen by both A and B with probability ${3\over10}\cdot{3\over10}$. The expected number of balls to be chosen twice therefore is $10\cdot{9\over100}=0.9$.
b) Any given ball is chosen by A, but not by B, with probability ${3\over10}\cdot{7\over10}$, and with the same probability it is chosen by B, but not by A. The expected number of balls to be chosen exactly once therefore is $10\cdot2\cdot{21\over100}=4.2$.