Fair and biased dice problem

Question

We have 2 dice - one is a fair die while the faces of the other die are 1, 1, 2, 2, 5, 5. We pick one of the two dice at random and throwing it gives a 2. What is the probability that we picked up the fair die?

Answer 1

For short, lets write BD= Biased dice, FD=Fair dice & Two= Threw a $2$ then \begin{eqnarray*} P(BD \text{ and } Two)=P(BD)\times P( Two \text{ on the Biased dice})=\frac{?}{?} \\ P(FD \text{ and } Two)=\underbrace{P(FD)}_{\frac{1}{2}}\times P( Two \text{ on the Fair dice})=\frac{?}{?} \\ P( Two)=\underbrace{P(BD \text{ and } Two)+P(FD \text{ and } Two)}_{\text{These events are mutually exclusive}} =\frac{?}{?} \\ \end{eqnarray*} Now use the conditional probability formula \begin{eqnarray*} P(FD \mid Two)= \frac{P(FD \text{ and } Two)}{P(Two)} = \frac{?}{?}. \end{eqnarray*}

$P(FD \mid Two)= \frac{\frac{1}{12}}{\frac{1}{4}} = \frac{1}{3}$