Q3
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Probability
└── Homework
└── W6
└── Q3.tex
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\begin{document}
\section*{Question 3}
\noindent Alice proposes to Bob the following bet. Alice tosses a fair coin n times, and computes the number of heads $X$. Bob tosses the coin $n+1$ times, and obtain $Y$ heads. Bob wins the bet if $Y > X$.
\bigskip
\begin{enumerate}[start=1,label={\bfseries Part \arabic*:},leftmargin=0in]
\bigskip\item Is the bet fair?
\subsection*{Solution}
As given, we can define the Bernoulli distribution with sample space $\Omega = \{\text{head},\text{tail}\}$ ($head$ denotes $drop\enspace headed$ and $tail$ denotes $drop\enspace tailed$) and with parameter $p = 0.5$ (by applying the principle of symmetry), then we can define the Binomial distribution of both $X$ and $Y$ as
\[
\begin{aligned}
X\sim &\text{Bin}(n,p)\\
Y\sim &\text{Bin}(n+1,p)
\end{aligned}
\]
Let as define $Z = Y - X$, then we have
\[Z\sim \text{Bin}(1,p)\]
Then we have the distribution of $Z$
\[
p(z) =
\begin{cases}
\begin{aligned}
0.5,\quad z = 0\\
0.5,\quad z = 1
\end{aligned}
\end{cases}
\]
That is
\[
\begin{aligned}
\bP(Y > X) &= \bP(Z > 0)\\
&= \frac{p(1)}{p(0) + p(1)}\\
&= \frac{1}{2}
\end{aligned}
\]
\subsection*{Answer}
\[\boxed{\text{Fair.}}\]
\bigskip\item Compute the answer for a general coin.
\subsection*{Solution}
Here we let $p_g$ denote a random variable in $(0,1)$, then we have the $Z$ in general case
\[Z_g\sim \text{Bin}(1,p_g)\]
Then we have the distribution of $Z_g$
\[
p_g(z) =
\begin{cases}
\begin{aligned}
1-p_g,\quad z = 0\\
p_g,\quad z = 1
\end{aligned}
\end{cases}
\]
By denoting $X$ and $Y$ in general case as $X_g$ and $Y_g$, we have
\[
\begin{aligned}
\bP(Y_g > X_g) &= \bP(Z_g > 0)\\
&= \frac{p_g(1)}{p_g(0) + p_g(1)}\\
&= p_g
\end{aligned}
\]
That is
\[
\text{Bet is}
\begin{cases}
\begin{aligned}
\text{Unfair, Alice tends to win more},&\quad p_g < 0.5\\
\text{Fair},&\quad p_g = 0.5\\
\text{Unfair, Bob tends to win more},&\quad p_g > 0.5
\end{aligned}
\end{cases}
\]
\subsection*{Answer}
\[\boxed{\text{Bet is}
\begin{cases}
\begin{aligned}
\text{Unfair, Alice tends to win more},&\quad p_g < 0.5\\
\text{Fair},&\quad p_g = 0.5\\
\text{Unfair, Bob tends to win more},&\quad p_g > 0.5
\end{aligned}
\end{cases}}\]
\end{enumerate}
\end{document}