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Chapter 3: Q. 3.1 (page 106)

Show that if P(A)>0, then

P(AB鈭�A)鈮�P(AB鈭�A鈭�B)

Short Answer

Expert verified

We proved that P(AB鈭�A)鈮�P(AB鈭�A鈭�B)by applying conditional probability asP(A)>0.

Step by step solution

01

Given Information

IfP(A)>0, We have to prove thatP(AB鈭�A)鈮�P(AB鈭�A鈭�B).

02

Explanation

AB鈯�(A鈭�B)鈬�AB鈭�(A鈭�B)=AB

The conditional probability P[AB鈭�(A鈭�B)]can be reduced to

P[AB鈭�(A鈭�B)]=P[AB鈭�(A鈭�B)]P(A鈭�B)=P(AB)P(A鈭�B)

Again, from set arithmetic

A鈯�A鈭�B鈬�P(A)鈮�P(A鈭�B)

Finally

P(AB鈭�A)=P(AB)P(A)鈮�P(AB)P(A鈭�B)=P[AB鈭�(A鈭�B)]

03

Final Answer

P(AB鈭�A)- is the percentage of Athat is in role="math" localid="1647853157806" AB

P[AB鈭�(A鈭�B)]- is the percentage of A鈭�Bthat is in AB.

And since A鈭�B is larger,P[AB鈭�(A鈭�B)]鈮�P(AB鈭�A).

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