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Chapter 2: Q. 2.12 (page 53)

Show that the probability that exactly one of the eventsEorFoccurs equals P(E)+P(F)鈭�2P(EF).

Short Answer

Expert verified

Note that the event in which only oneE,Foccurs isP(E鈭�F)(EF)c

Use Axiom3for (E鈭�F)(EF)c&E鈭�Fand Proposition4.3.

Step by step solution

01

Given Information.

The probability that exactly one of the eventsEorFoccurs equalsP(E)+P(F)鈭�2P(EF)..

02

Explanation.

For any events EandF:

P(E鈭�F)(EF)c=P(E)+P(F)-2P(EF)

Where the left-hand side is the event where precisely one event E&Foccurs.

Because(E鈭�F)(EF)cis an event that occurs ifEor Foccur and not both Eandlocalid="1649246913116" Foccur.

Firstly note that

(E鈭�F)(EF)c鈭�EF=鈭�(E鈭�F)(EF)c鈭�EF=E鈭�F

Therefore

P(E鈭�F)=P(E鈭�F)(EF)c+P(EF)P(E鈭�F)(EF)c=P(E鈭�F)-P(EF)=P(E)+P(F)-P(EF)-P(EF)=P(E)+P(F)-2P(EF)

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Most popular questions from this chapter

Balls are randomly removed from an urn initially containing 20red and 10blue balls. What is the probability that all of the red balls are removed before all of the blue ones have been removed?

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A system is composed of 5components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector (x1,x2,x3,x4,x5), where xiis equal to 1if component iis working and is equal to 0if component iis failed.

(a) How many outcomes are in the sample space of this experiment?

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