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 time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what i

Explanation

Let`E_1`= event of selecting a two headed coin.
`E_1` = event of selcting a baised coin, which shows 75 % times Head
`E_3` = event of selecting a unbaised coin.
A = event that tossed coin shows head.
`" " therefore P(E_1)= P(E_2) = P(E_3) = (1)/(3)`
`P(A//E_1)`= P(coin showing head given that uit is two headed coin)
`" "=1`
`P(A//E_2)`= P( coin showing head given that it is a biased coin)
`" "=(75)/(100) = (3)/(4)`
`P(A//(E_(3))`=P(coin showing head given that it is unbaised coin)
`" " = (1)/(2)`
By Bayes theorem
P(gettingtwo headed coin when it is known that it shows Head)
`P(E_1//A) = (P(E_1)P(A//E_1))/(P(E_1)P(A//E_1)+ P(E_2)P(A//E_2)+P(E_1)P(A//E_3))`
`= ((1)/(3)xx 1)/((1)/(3)xx1 + (1)/(3)xx (3)/(4) + (1)/(3) xx (1)/(2)) = ((1)/(3) xx 1)/((1)/(3)(1+(3)/(4)+(1)/(2))) = ((1)/(3))/((1)/(3)xx(9)/(4))= (4)/(9)`
Required probability = `(4)/(9)`