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Question

The contents of three urns are as given below:
1 white, 2 black, 3 red balls
2 white, 1 black, 1 red balls
4 white, 5 black, 3 red balls
One urn is chosen at random and two balls are drawn. These happen to be white and red. What is the probability that they came from urn 2

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Solution

I1white2black3redII2white1black1redIII4white5black3redLetE1,E2,E3andAdenotethefollowingevent:E1=UrnIischoosenE2=UrnIIischoosenE3=UrnIIIischoosenA=Twoballsaredrawnatrandom&whiteandred.ProbabilityofUrnP(E1)=p(E2)=P(E3)=13case(I)choosingUrn(I)Totalnumberofballs=6andhavetochoose=2balls(1)Numberofwhiteball=1Numberofblackball=2Numberofredball=3P(A/E1)=1c1.3c16c2[ncr=|!n|!n.|!nn]=1×315=15Similarly,P(A/E2)=2c1.1c14c2=2×16=13P(A/E3)=4c1.3c112c2=4×366=211usingBaystheorem.P(E2/A)=P(E2).P(A/E2)P(E1)P(A/E1)+P(E2)P(A/E2)+P(E3)P(A/E3)13×1313×13+13×13+13×21119115+19+2331913(13+15+211)13(55+33+30)1651311816513×165118P(E2/A)=55118Ans.

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