exercise01.14.tex

\paragraph{Exercise 1.14} I am playing in a racquetball tournament, and I am up
against a player I have watched but never played before. I consider three
possibilities for my prior model: we are equally talented, and each of us is equally
likely to win each game; I am slightly better, and therefore I win each game
independently with probability 0.6; or he is slightly better, and thus he wins
each game independently with probability 0.6. Before we play, I think that each
of these three possibilities is equally likely. In our match we play until one
player wins three games. I win the second game, but he wins the first, third, and
fourth. After this match, in my posterior model, with what probability should I
believe that my opponent is slightly better than I am?
Let $B$ be the event that I am slightly better than my opponent, $E$ the event that
we are equally talented, and $W$ the event that my opponent is slightly better than
I am. $B, E$ and $W$ are mutually disjoint events. Let LWLL be the event that I
win the second game and my opponent wins th first, third and fourth game. \\
We will apply Bayes' Law to compute the probability that my opponent is slightly
better than I am under the condition that LWLL occurs.
\begin{align*}
  \pr(W|\text{LWLL})
    &= \frac{\pr(\text{LWLL}|W) \cdot \pr(W)}{\pr(\text{LWLL}|B) \cdot \pr(B) + \pr(\text{LWLL}|E) \cdot \pr(E) + \pr(\text{LWLL}|W) \cdot \pr(W)} \\
    &= \frac{\left(\frac{3}{5}\right)^3 \cdot \frac{2}{5} \cdot \frac{1}{3}} {\left( \frac{2}{5} \right)^3 \cdot \frac{3}{5} \cdot \frac{1}{3} + \left( \frac{1}{2} \right) ^4 \cdot \frac{1}{3} + \left(\frac{3}{5}\right)^3 \cdot \frac{2}{5} \cdot \frac{1}{3}} \\
    &= \frac{864}{1873} \\
    &\approx 0.461.
\end{align*}
Hence, in my posterior model, the probability that my opponent is slightly better
than I am is approximately 0.461.