# Coin toss probability

Three coins but into hat. Coin 1 shows heads on both sides, coin 2 is a fair coin, and coin 3 has a .75 probability of landing heads...one coin is randomly selected and flipped. The coin lands heads side up. What is the probability that the other side is also heads? ... My friend and I are debating the answer to this question. One of us thinks it is 12/27, and the other 1/3. Who is correct?

So we want to calculate the conditional probability of $P(A|B)$.
According to Baye's theorem we get: $$P(A|B)=\frac{P(A)P(B|A)}{P(B)}$$ $$P(A)=\frac{1}{3}$$ $$P(B)=\frac{1+0.5+0.75}{3}=\frac{3}{4}$$ $$P(B|A)=1$$ $$\therefore P(A|B)=\frac{\frac{1}{3}}{\frac{3}{4}}=\frac{4}{9}(=\frac{12}{27})$$