Suppose that A and B each randomly, and independently, choose 3 of 10 objects. Find the expected number of objects chosen by both A and B.
solution: Let $X$ be the number of objects chosen by both A and B. For $1≤i≤10$, let $X_i=1$ if object $i$ is chosen by A and B, else $0$ otherwise
We find $E[X_i]=0⋅P(X_i=0)+1⋅P(X_i=1)=P(X_i=1)=9/100$.
By the linearity of expectation, $E[X]=10⋅E[Xi]=0.9$
I understand this approach but I can't get the same answer by multiplying the probability of A and B having 1,2,3 objects in common by the values 1,2,3 and adding them, why is this approach concerned wrong ? I assumed x is the number of objects chosen by both thus it takes the values 1,2,3