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Chapter 3: Q. 91 (page 224)

Yand Zare independent events.

a. Rewrite the basic Addition Rule P(YORZ)=P(Y)+P(Z)-P(YANDZ)using the information that Y and Z are independent events.

b. Use the rewritten rule to findP(Z) if P(YORZ)=0.71and PY=0.42

Short Answer

Expert verified

(a) P(YORZ)=P(Y)+P(Z)-P(Y)P(Z).

(b)PZ=0.5.

Step by step solution

01

Given information (part a)

Y andZ are independent events.

02

Explanation (part a)

The addition rule states that

PYorZ=PY+PZ-PYandZ

Also, given that Yand Z are independent events

Thus, we have PYandZ=PYPZ

Substituting the value in the above rule, we get

PYorZ=PY+PZ-PYPZ

03

Given information (part b)

Y and Z are independent events.PYorZ=0.71andPY=0.42

04

Explanation (part b)

The addition rule states that

P(YORZ)=P(Y)+P(Z)-P(YANDZ)

Also, given that Y andZ are independent events

Thus, we have PYandZ=PYPZ

Substituting the value in the above rule, we get

P(Y鈭�Z)=P(Y)+P(Z)-P(Y)P(Z)P(Y鈭�Z)=P(Y)+P(Z)-P(Y)P(Z)

Now, we have

PYorZ=0.71PY=0.42

Substituting the values, we get

0.71=0.42+P(Z)-(0.42脳P(Z))0.71=0.42+[P(Z)(1-0.42)]0.71=0.42+0.58P(Z)P(Z)=0.71-0.420.58PZ=0.5

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