# There are three coins. One is a two-headed coin, another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows heads, what is the probability that it is a two- headed coin?

Given there are 3 coins

Let,

E1 = coin is two headed

E2 = biased coin

E3 = unbiased coin

Here

$P(E_1)=P(E_2)=P(E_3)=\frac{1}{3} \\ then , \: P(\frac{A}{E_1})=1 \: \: \: \: \: P\left ( \frac{A}{E_2} \right )= \frac{1}{2} \\ P\left ( \frac{A}{E_2} \right )= \frac{75}{100}=\frac{3}{4}(given)$

Now the probability of 2 headed coin

$P(E_1/A)= \frac{P(E_1)P(A/E_1)}{P(E_1)\times P(A/E_1)+P(E_2)\times P(A/E_2)+P(E_3)\times P(A/E_3)}\\ \frac{\frac{1}{3}\times 1}{\frac{1}{3}\times 1 +\frac{1}{3}\times \frac{3}{4}+ \frac{1}{3}\times \frac{1}{2}}=\frac{1}{\frac{4+3+2}{4}}= \frac {4 }{9}$

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