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Chapter 4: Q.4.1 (page 169)

There are N distinct types of coupons, and each time one is obtained it will, independently of past choices, be of type i with probability Pi, i = 1, ... , N. Let T denote the number one need select to obtain at least one of each type. Compute P{T = n}.

Short Answer

Expert verified

In the given information the answer isP(T=n)=P(T>n-1)-P(T>n)

Step by step solution

01

Step 1:Given Information

Consider eventAi,which states that we havent picked a coupon of type i in our sample i=1.....N so T>nis equal to the information that is satisfied some of the event Ai.ie,

P(T>n)=P∪i=1NAi

02

Calculation

P∪i=1NAi=∑i1PAi1-∑i1<i2PAi1,Ai2+∑i1<i2<i3PAi1,Ai2,Ai3

PAi1=1-pi1n

PAi1,Ai2=1-pi1+pi2n

PAi1,Ai2,…,Aij=1-pi1+pi2+⋯+pijn

The required probability T=n can be obtained asP(T=n)=P(T>n-1)-P(T>n)

03

Step 3:Final Answer

The final answer isP(T=n)=P(T>n-1)-P(T>n)

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