There are three coins. One is a two-headed coin (having head on b… - Vidyadip

Question

There are three coins. One is a two$-$headed coin $($having head on both faces$),$ another is a biased coin that comes up heads $75\%$ of the times and third is also a biased coin that comes up tails $40\%$ of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two$-$headed coin?

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Answer

Let event $E_1:$ choosing first $($two headed$)$ coin
$E_2:$ choosing $2^{nd} ($biased$)$ coin
$E_3:$ choosing $3^{rd} ($biased$)$ coin
$\therefore\text{ P(E}_{1}) = \text{P(E}_{2}) = \text{P(E}_{3}) = \frac{1}{3}$
$A:$ The coin showing heads.
$\therefore\text{ P(A/E}_{1}) = 1, \text{P(A/E}_{2}) = \frac{75}{100} = \frac{3}{4},\text{ P(A/E}_{3}) = \frac{60}{100} =\frac{3}{5}$
$P(E_1/A) = \frac{\frac{1}{3}.1}{\frac{1}{3}.1 + \frac{1}{3}.\frac{3}{4} + \frac{1}{3}.\frac{3}{5}}$
$ =\frac{20}{47}.$

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