Consider two coins: a regular fair coin and a two-headed one (i.e., both sides are head). Pick a coin at random and toss it. If its top side shows a head, what is the probability that the other side is also a head?
My understanding is that first we will calculate the probability of head using partitions. The probability of head given any outcome of the coin comes out to be 0.75. Then, which probability we have to calculate further to answer this question. Do we have to apply conditional probability by finding the probability of two-headed coin given that head is selected. My answer comes out to be 0.6667. Kindly help me figure it out.
Yes the probability that the coin is the double header when given that it shows heads is $2/3$.$$\begin{align}\mathsf P(D\mid H)&=\dfrac{\mathsf P(H\mid D)\,\mathsf P(D)}{\mathsf P(H\mid D)\,\mathsf P(D)+\mathsf P(H\mid S)\,\mathsf P(S)}\\[1ex]&=\dfrac{1/2}{1/2+1/4}\\[1ex]&=\dfrac{2}{3}\end{align}$$Each individual face has the same chance to show and two from the three heads are on the double-headed coin.