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Chapter 4: Q.4.10 (page 173)

An urn containsnballs numbered 1through n. If you withdraw mballs randomly in sequence, each time replacing the ball selected previously, findP{X=k},k=1,...,m

where Xis the maximum of the mchosen numbers.

Hint: First find P{X鈮�k}.

Short Answer

Expert verified

P(X鈮�k)=knm

P(X=k)=knm-k-1nm

Step by step solution

01

Step 1: Given information

An urn contains n balls numbered1 through n. If you withdraw m balls randomly in sequence, each time replacing the ball selected previously.

02

Step 2: Explanation

For the beginning, we will find P(X鈮�k). Observe that in total, there existnmsequences of drawn balls. If X鈮�k, that means all balls in the sequence that has been drawn must be less or equal to k. There exist kmof these sequences. Hence

P(X鈮�k)=kmnm=knm

Now, we have that

P(X=k)=P(X鈮�k)-P(X鈮�k-1)=knm-k-1nm

03

Step 3:Final answer

P(X鈮�k)=knm

P(X=k)=knm-k-1nm

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