orP(F∩E) =P(E)P(F),which agrees with the definition of independence we gave before.Let us give two more examples.An example: Suppose an urn holds 5 red balls and 7 green balls.Youdraw two balls without replacement. What is the probability the second ballis red?Answer. LetAbe the event that the first ball is red,Bthat the secondball is, and we wantP(B). ThenP(B) =P(A∩B) +P(Ac∩B) =P(B|A)P(A) +P(B|Ac)P(Ac).The probability ofAis512and the probability forAcis712. Given that thefirst ball is red, there are now 4 red balls and 7 green, soP(B|A) =411.Similarly,P(B|Ac) =511. ThereforeP(B) =411·512+511·712=512,which is what one would expect.An example. This is known as the Monty Hall problem after the host ofthe TV show of the 60’s calledLet’s Make a Deal.There are three doors, behind one a nice car, behind each of the othertwo a bale of straw. You choose a door. Then Monty Hall opens one of theother doors, which shows a bale of straw. He gives you the opportunity of
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