Q. 6.     There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75^{o}/_{o} of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?

Answers (1)

Given :   A :  chossing a two headed coin

              B :  chossing a biased coin

              C : chossing a unbiased coin 

P(A)=P(B)=P(C)=\frac{1}{3}

             D : event that coin tossed show head.

         P(D|A)=1

     Biased coin that comes up heads 75^{o}/_{o} of the time.

       P(D|B)=\frac{75}{100}=\frac{3}{4}

     P(D|C)=\frac{1}{2}

P(B|D)=\frac{P(B).P(D|B)}{P(B).P(D|B)+P(A).P(D|A)+P(C).P(D|C)}

P(B|D)=\frac{\frac{1}{3}\times 1}{\frac{1}{3}\times 1+\frac{1}{3}\times \frac{3}{4}+\frac{1}{3}\times \frac{1}{2}}

P(B|D)=\frac{\frac{1}{3}}{\frac{1}{3}+\frac{1}{4}+\frac{1}{6}}

P(B|D)=\frac{\frac{1}{3}}{\frac{9}{12}}

P(B|D)={\frac{1\times 12}{3\times 9}}

P(B|D)={\frac{4}{9}}

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