# Tossing 2 Coins, Fair coin and a two headed coin

There are two coins. One is two headed coin, the other one is a fair coin. A coin is selected randomly and flipped.

a) what is the probability of selecting a fair coin?

b) what is the probability of head from the flipped coin?

c) what is the probability of selecting a fair coin and having head on the top?

So A and B are relatively simple, because A is $$1/2$$ and for B its $$3/4$$ but I am concerned with C

Shouldn't be the chance of having a fair coin $$+$$ head is basically $$1/4$$? Correct me if I am wrong but

If I want to pick a fair coin, the probability is $$0.5$$, lets say we picked it, and now we are going to throw it, isn't also chance between head and tail is $$0.5$$? Which effectively means that getting a head from a fair coin

the intersection between them is $$0.5\times0.5$$ which equals $$0.25$$?

• Is the answer to the c part, 1/3? Jul 30, 2022 at 14:53
• @Arsenic $P(A\land B)=P(A|B)P(B)$ so the probability of getting the fair coin and getting a head is the probability of getting a head, given that we have a fair coin ($\frac{1}{2}$), times the probability of getting a fair coin which is also $\frac{1}{2}$ so we get $\frac{1}{4}$. If we chose a coin, flipped it and got a head then the probability that the head came from the fair coin is $\frac{1}{3}$. Jul 30, 2022 at 15:02
• @Arsenic The prof didn't give us any answer or key sol I am sorry Jul 30, 2022 at 15:02
• @JohnDouma Yep! I presumed that the answer OP posted in the question was marked incorrect, so could only see conditional probability being a reason- which would have meant that the question wasn't framed correctly. Jul 30, 2022 at 15:07

Let $$F=$$ "select a fair coin", and $$H=$$ "appear head on top"
$$P(F\cap H)=P(H|F)\cdot P(F)=\frac{1}2\cdot\frac{1}2=\frac{1}4$$
• You are most welcome! If the answer helps, please $\checkmark$ my answer :) @Eimon Jul 30, 2022 at 15:13