Suppose that 10 balls are put into 5 boxes, with each ball independently being put in box i with probability
p_(i), sum_(i=1)^(5)p_(i)=1
Find the expected number of boxes that do not have any balls.

1 Answers

Best Answer

Define indicator random variables I_(i), i=1,...., 5 that marks whether i^(th) box is empty or not. Observe that
P(I_(i)=1)=(1-p_(i))^(10)
since we have that i^(th) box is empty if and only if all the balls have been put in remaining boxes and each ball is put in some other box rather that i^(th) box with probability 1-p_(i). Hence, the number of empty boxes can be expressed as N= sum_(i=1)^(5) I_(i), so, using the linearity of expectation, we get
E(N)= sum_(i=1)^(5)E(I_(i))= sum_(i=1)^(5)P(I_(i)=1)= sum_(i=1)^(5)(1-p_(i))^(10)