# Elements of Probability coin flip

A purse contains three fair coins, two double headed and one double tailed. A coin is chosen randomly . What is the probability of picking a double headed coin given that it landed on heads?

• Unless I am missing something, the double tailed coin cannot land on heads so ... – gandalf61 Jul 25 at 14:16
• @gandalf61 The purse contains six coins in total. – Magma Jul 25 at 14:22
• @Magma Oh, I see. Three HT coins and two HH coins and one TT coin. That was not how I read the question, but it does make it less trivial ! – gandalf61 Jul 25 at 15:11
• @gandalf61 You are right the double tailed coin cannot land on head. So there are two more coins that are double headed. The way you read it was correct indeed. I used the baye's theorem and two tuples of a coin and its outcome. My answer was 1. I felt it was a bit interesting if my calculation is indeed right. Although intuitively anyone will say1 without doing the calculation . – Nazifa Taha Jul 25 at 16:26

Let $$A$$ denote the event of picking a double-headed coin, and $$B$$ denote the event of picking a head. We wish to calculate $$\mathbb{P}(A|B)$$.
By Bayes, $$\mathbb{P}(A|B) = \frac{\mathbb{P}(B|A)\mathbb{P}(A)}{\mathbb{P}(B)}$$.
Clearly $$\mathbb{P}(B|A)=1$$, $$\mathbb{P}(A) = \frac{2}{6} = \frac{1}{3}$$ and $$\mathbb{P}(B) = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{6} + \frac{1}{6} + 0 = \frac{7}{12}$$.
Therefore your answer is $$$$\boxed{\frac{4}{7}}$$$$