exercise01.10.tex

\paragraph{Exercise 1.10} Assume that we have a fair coin and a two-headed coin.
We choose one of the two coins randomly with equal probability and flip it. Given
that the flip was heads, what is the probability that we flipped the two-headed coin? \\
Let $F$ be the event that we choose the fair coin and $U$ the event that we choose
the two-headed coin. Let $H$ be the event that the choosen coin comes up heads
and $T$ the event that it comes up tails. \\
We are interested in the conditional probability $\pr(U|H)$. Applying Bayes' Law
one has
\[ \pr(U|H)
    = \frac{\pr(U \cap H)}{\pr(H)}
    =\frac{\pr(H|U) \cdot \pr(U)}{\pr(H|U) \cdot \pr(U) + \pr(H|F) \cdot \pr(F)}
    =\frac{1 \cdot \frac{1}{2}}{1 \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2}}
    = \frac{2}{3}.
\]