
\problemdata     
{S01-PS9-7}           
{probability-conditiona, independence}              
{} 		
{S01-PS9}
{S01 Karger}


\begin{problem}



\textbf{The Flippant Juror}

A three-person jury has two responsible members, each of whom
independently has a probability $p> 1/2$ of making the correct decision.
The third juror flips a 50-50 coin for each decision.  So a one-person
jury with a responsible juror has probability $p$ of making the correct
decision.  Suppose the three-person jury decides by majority.  Which jury
has the better probability of making the correct decision, a
one-repsonsible-person jury or the three-person jury?

\solution{

Suppose the members of the 3-person jury are Socrates, Plato, and
Homer Simpson.  Socrates and Plato have both independently a probability
$p$ of making the right choice, and Homer just flips a coin.
Here are the events in which case the 3-person jury could be right:
\begin{itemize}
\item
Socrates and Plato are both right, (in which case what Homer says is irrelevant, as the
majority rules): probability of this event is  $p^2$;

\item
Socrates is right, Plato is wrong, and Homer is right: probability of this event is $\frac{1}{2}
p (1-p)$;

\item
Socrates is wrong, Plato is right, and Homer is right: probability of this event is $\frac{1}{2}
p (1-p)$.
\end{itemize}

These events are disjoint, so the probability that one of them occurs
is the sum $p^2 + p(1-p) = p$.  Thus the 3-person jury and the
1-person jury have the {\it same} probability of making the correct
decision.

}

\end{problem}

