5 Suppose we put r balls at random into 10 boxes such that a

(5) Suppose we put r balls at random into 10 boxes such that all the ior assignments of balls into boxes are equally likely. Let A be the event that the ith box is empty. (a) Compute P [A1] and P [A1 intersection A2]. (b) What is the probability that there is at least one empty box? (c) Show that Sigma (-1)^k (10-k)^r =

Solution

we put r balls into 10 boxes at random. Hence the events are independent and equally likely.

Let p be the probability for a success i.e.

Each ball goes to 1 box has a prob = 0.10

p =0.10 q =0.90 n =r

P(Ai) = P(ith box is empty) = q^r = (0.9)^r

P(A1IntA2) = (0.9)^2r (as the events are indepdendent)

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Let X be the no of boxes empty.

X can be 0,1,2.......r