probability

Question

Consider four coins, three of which are fair, that is they have heads on one side and tails on the other and both are equally likely to occur in a toss. The fourth coin has tails on both sides. Given that one coin amongst the four is picked at random and is tossed, and the outcome is seen to be tail, what is the probability that its other side is heads

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https://gateoverflow.in/103424/probability

Answer 1

X₁ => We choosed fair coin

X₂ => We choosed biased coin

X3 => we got tails

So we need to fid P(X₁ | X₃ ) that means if tails come then what is the probability that the coin is fair

so P(X₃ | X₁ ) =  1/2

and P(X₃ | X₂ ) = 1

P(X₁) = 3/4

P(X₂) = 1/4

P(X₃) = P(X₃ | X₁ )*P(X₁) + P(X₃ | X₂ )*P(X₂) = $\frac{1}{2}*\frac{3}{4}+1*\frac{1}{4}=\frac{5}{8}$

P(X₃ | X₁ ) * P(X₁) = P(X₁ | X₃) * P(X₃)

so  P(X₁ | X₃) = $\frac{\frac{1}{2}*\frac{3}{4}}{\frac{5}{8}}=\frac{3}{5}$