There are three coins. One is two headed coin, 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 choosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
Given:
Coin 1 is two heads, coin2 is biased and coin 3 is unbiased
Let us assume U1, U2, U3 and A be the events as follows:
U1 = choosing coin 1
U2 = choosing coin 2
U3 = choosing coin 3
A = getting heads
We know that each coin is most likely to choose. So, probability of choosing a coin will be same for every coin.
⇒ 
⇒ 
⇒ 
From the problem
⇒ P(A|U1) = P(getting heads on tossing coin 1)
⇒ 
⇒ P(A|U2) = P(getting heads on tossing coin 2)
⇒ 
⇒ P(A|U3) = P(getting heads on tossing coin 3)
⇒ 
Now we find
P(U1|A) = P(The coin tossed to get head is Coin 1)
Using Baye’s theorem:
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
∴ The required probability is 
.