Answer: The probability that it was the two-headed coin is
.
Step-by-step explanation:
Let we consider the events as
A = A two-headed coin is tossed
B = A fair coin is tossed
C = biased coin is tossed
H= Results heads after toss
As per given , total coins =3
So , ![P(A)=P(B)=P(C)=\dfrac{1}{3}](https://tex.z-dn.net/?f=P%28A%29%3DP%28B%29%3DP%28C%29%3D%5Cdfrac%7B1%7D%7B3%7D)
Probability that coins show heads :
![P(H|A)=1\\P(H|B)=\dfrac{1}{2}\\P(H|C)=0.75](https://tex.z-dn.net/?f=P%28H%7CA%29%3D1%5C%5CP%28H%7CB%29%3D%5Cdfrac%7B1%7D%7B2%7D%5C%5CP%28H%7CC%29%3D0.75)
By Bayes theorem ,we have
![P(A|H)=\dfrac{P(H|A)}{P(A)P(A|H)+P(B)P(B|H)+P(C)P(C|H)}\\\\=\dfrac{1(\dfrac{1}{3})}{1(\dfrac{1}{3})+\dfrac{1}{3}(\dfrac{1}{2})+\dfrac{1}{3}(0.75)}](https://tex.z-dn.net/?f=P%28A%7CH%29%3D%5Cdfrac%7BP%28H%7CA%29%7D%7BP%28A%29P%28A%7CH%29%2BP%28B%29P%28B%7CH%29%2BP%28C%29P%28C%7CH%29%7D%5C%5C%5C%5C%3D%5Cdfrac%7B1%28%5Cdfrac%7B1%7D%7B3%7D%29%7D%7B1%28%5Cdfrac%7B1%7D%7B3%7D%29%2B%5Cdfrac%7B1%7D%7B3%7D%28%5Cdfrac%7B1%7D%7B2%7D%29%2B%5Cdfrac%7B1%7D%7B3%7D%280.75%29%7D)
Simplify , we get ![\dfrac{4}{9}](https://tex.z-dn.net/?f=%5Cdfrac%7B4%7D%7B9%7D)
Hence, the probability that it was the two-headed coin is
.