# A bag contains two coins, one biased and the other unbiased. When tossed, the biased coin has a 60% chance of showing heads. One of the coins is selected at random and on tossing it shows tails. What is the probability it was an unbiased coin ?

$\\ \mathrm{E}_{1}: Unbiased coin is tossed$

$\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{1}{2}$

$\mathrm{E}_{2}: Biased coin is tossed$

$\mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{1}{2}$

$A: Coin tossed shows the tail$

$\mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{1}\right)=\frac{1}{2}$

$\mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{2}\right)=1-\frac{60}{100} = \frac{2}{5}$

$P\left(\frac{E_{1} }{A}\right) =\frac{P\left(E_{1}\right) \cdot P\left(\frac{A}{E_{1}}\right)}{P\left(E_{1}\right) \cdot P\left(\frac{A}{E_{1}}\right)+P\left(E_{2}\right) \cdot P\left(\frac{A}{E_{2}}\right)}$

$P\left(\frac{E_{1} }{A}\right) =\frac{\frac{1}{2} \times \frac{1}{2}}{\frac{1}{2} \times \frac{1}{2}+\frac{1}{2} \times \frac{2}{5}}=\frac{5}{9}$

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