Suppose you have two coins which appear identical in your pocket. You know that, one is fair and one is 2 headed. If you take one out toss it and get a head, what is the probability that it was a fair coin?

Let ${E_1} =$ Event that fair coin is drawn

${E_2} =$ Event that 2 headed coin is drawn

$E =$ Event that tossed coin get a head

$\therefore$ $P\left( {{E_1}} \right) = 1/2,$ $P\left( {{E_2}} \right) = 1/2,$ $P\left( {E/{E_1}} \right) = 1/2$ and $P\left( {E/{E_2}} \right) = 1$

Now, using Baye's theorem $P\left( {{E_1}/E} \right) = \frac{{P\left( {{E_1}} \right) \cdot P\left( {E/{E_1}} \right)}}{{P\left( {{E_1}} \right) \cdot P\left( {E/{E_1}} \right) + P\left( {{E_2}} \right) \cdot P\left( {E/{E_2}} \right)}}$

$= \frac{{\frac{1}{2} \cdot \frac{1}{2}}}{{\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot 1}} = \frac{{\frac{1}{4}}}{{\frac{1}{4} + \frac{1}{2}}} = \frac{{\frac{1}{4}}}{{\frac{3}{4}}} = \frac{1}{3}$