Identifying an Unfair Coin With a Flip

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  
\[\frac{1}{2} \times \frac{3}{4}=\frac{3}{8}\]
.
\[\begin{equation} \begin{aligned} P(Heads \: on \: the \: Unfair : One) &=\frac{P(Heads \; on \: the \: Unfair \: One \cap Tails \: on \: the \: Fair \: One)}{P(Different \: Side \: Up \: On \: Different \: Coins)} \\ &= \frac{3/8}{1/8+3/8} \\ &= \frac{3}{4} \end{aligned} \end{equation}\]

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