A box contains three coins. One is a fair coin, the second is an unfair-tail coi
ID: 3277017 • Letter: A
Question
A box contains three coins. One is a fair coin, the second is an unfair-tail coin with a tail on each side, and the third is an unfair-head coin with a head on each side. a. Suppose a coin is selected at random and that when tossed, a tail is obtained. What is the probability that it is a fair coin? b. Now suppose that the same coin is tossed again and another tail is obtained. Now what is the probability that it is a fair coin? Assume that the two tosses are conditionally independent given the knowledge of which coin was tossed.
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Explanation / Answer
here let probability of fair coin is P(F); that of unfair with both sides tails =P(UT) and of unfair with both sides heads=P(UH).
a) probability of tail =P(T) =P(F)*P(T|F)+P(F)*P(T|UH)+P(T)*P(T|UT)=(1/3)*(1/2)+(1/3)*0+(1/3)*(1) =1/2
therfore probability that it is a fair coin given a tail is obtained =P(F)*P(T|F)/P(T) =(1/3)*(1/2)/(1/2) =1/3
b) probability of two consecutive tail =P(TT) =P(F)*P(TT|F)+P(F)*P(TT|UH)+P(TT)*P(T|UT)
=(!/3)*(1/2)*(1/2)+(1/3)*0*0+(1/3)*(1)*(1) =5/12
therefore probability that it is a fair coin; given two consecutive tail = P(F)*P(TT|F)/P(TT) =(1/3)*(1/2)*(1/2)/(5/12)
=1/5
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