Question

There are three coins. One is 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 is the probability that it was the two headed coin?

Answer 1

Let E₁: The coin is head on both sides

E₂: the coin is biased

E₃: the coin is unbiased

E: The coin tossed shows the head

P(E₁) = P(E₂) = P(E₃) = `1/3`

P(E|E₁) = 1

P(E|E₂) = 75% = `75/100 = 3/4`

P(E|E₃) = `1/2`

Intended process

P(E₁|E) = `(P(E_1) xx P(E|E_1))/(P(E_1) xx P(E|E_1) + P(E_2) xx P(E|E_2) + P(E_3) xx P(E|E_3)`

= `(1/3 xx 1)/(1/3 xx 1 + 1/3 xx 3/4 + 1/3 xx 1/2)`

= `1/(1 + 3/4 + 1/2)`

= `4/(4 + 3 + 2)`

= `4/9`