You have two coins that look the same. One of them is not - it has a
\[\frac{3}{4}\]
chance of getting heads if flipped.Unfortunately, you don't know which is which. You decide to flip them until you get a head on one coin and a tail on other, then take the one showing heads as the unfair coin.
What is the probability of picking the unfair coin?
We can construct the probability table below.
| Fair | H(1/2) | H(1/2) | T(1/2) | T(1/2) |
| Unfair | H(3/4) | T(1/4) | H(3/4) | T(3/4) |
The probability of flipping a head on the fair coin and tails on the unfair coin is
\[\frac{1}{2} \times \frac{1}{4}=\frac{1}{8}\]
.The probability of flipping tails on the fair coin and a head on the unfair coin is
\[\frac{1}{2} \times \frac{3}{4}=\frac{3}{8}\]
.\[\begin{equation} \begin{aligned} P(Heads \: on \: the \: Unfair : One) &=\frac{P(Heads \; on \: the \: Unfair \: One \cap Tails \: on \: the \: Fair \: One)}{P(Different \: Side \: Up \: On \: Different \: Coins)} \\ &= \frac{3/8}{1/8+3/8} \\ &= \frac{3}{4} \end{aligned} \end{equation}\]