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Question

A man speaks truth 3 times out of 5. He says that in tossing 6 coins, two tails appear so what is the probability that this event is actually true ?


Solution

Let E represents the statement of a person , Now $$ S_{1}$$ is the event of getting heads and $$ S_{2}$$ is the event of not getting 2 heads on tossing 6 coins .
$$ \therefore  P(S_{1}) = ^{6}C_{2} \left ( \dfrac{1}{2} \right )^{6-2} \left ( \dfrac{1}{2} \right )^{2} $$
$$ = ^{6}C_{2} \left ( \dfrac{1}{2} \right )^{6} = \dfrac{15}{64}$$
$$  P(S_{2}) =  1 - P(S_{1}) $$
$$ =  1 - \dfrac{15}{64} = \dfrac{49}{64} $$
$$ = \dfrac{\frac{15}{64}\times \dfrac{3}{5}}{\frac{15}{64}\times\dfrac{3}{5} + \dfrac{49}{64}\times \dfrac{2}{5}} = \dfrac{45}{45 + 98} = \dfrac{45}{143}$$

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