\problemdata                                                             
        {Nicole and OJ car crash} % latex-friendly label for the prob.
        {Conditional Probability} % The topic(s) of the problem content.  
        {?} % Source (if known)                                          
        {S01 Tutorial 11, 4} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
Mr. Simpson has been involved in a car crash with Ms. Nicole, who then decided to sue Mr. Simpson for damages, including the loss of her antique Volkswagen Beattle and physical injuries she sustained when her hot coffee was spilled on her lap during the crash. There is very little incriminating evidence against Mr. Simpson, but Ms. Nicole's lawyers maintain that he is very likely to have been responsible for the car crash because it is widely known that Mr. Simpson routinely talks on his cell phone while driving (implying that the car crash must have taken place while he was talking on the phone, which increases the likelihood that \emph{he} was responsible for the crash).  

Defense attorney Dershowitz observed that the fraction of drivers who talk on their cell phone while driving and are involved in a car crash while doing so is $1/2000$.

Since the probability that someone who ``talks and drives" ends up causing an accident is so
low, Dershowitz maintains that the fact that Mr. Simpson talks on his cell phone while driving should be excluded from the evidence.  

We will develop a counter-argument to Dershowitz', by conditioning the probability
on the event that there was a car crash.

We have the following statistical information concerning cell phones and car crashes, for the period of one week:
\begin{center}
\begin{tabular}{|l|c|}
\hline
drivers & 125M\\
drivers with cell phones & 50M\\
drivers who talk and drive & 3M\\
drivers who caused crashes& 6K\\
drivers who caused crashes while talking and driving& 1.5K\\
\hline
\end{tabular}
\end{center}
The number of drivers who talk and drive includes those who caused a crash while talking and driving.  Define the events
\begin{eqnarray*}
H & ::= & \mbox{driver who causes a crash while talking and driving} \\
M & ::= & \mbox{driver who causes a crash} \\
B & ::= & \mbox{driver who routinely talks and drives}
\end{eqnarray*}

\begin{problemparts}

\problempart Dershowitz' statistical observation can be stated technically as
\[\prob{H \mid B} = 1/2000.\]
Prove it.

\solution{
\[
\prob{H \mid B}
= \frac{\prob{H \intersect B}}{\prob{B}}
=\frac{\prob{H}}{\prob{B}}=\frac{1500/125M}{3M/125M}
=\frac{1500}{3M}=1/2000.
\]}

\problempart Calculate $\prob{H \mid M}$, i.e., the probability that a driver who caused a car crash was ``talking and driving" during the crash.

\solution{
\[\prob{H \mid M}
= \frac{\prob{H \intersect M}}{\prob{M}}
=\frac{\prob{H}}{\prob{M}}=\frac{1500/125M}{6000/125M}
= \frac{1500}{6000}=1/4.
\]}

\problempart Prove
\begin{equation}
\label{ABA-B}
\prob{C \mid A}  =  \prob{C \intersect D \mid A} + \prob{C - D \mid A}
\end{equation}
for arbitrary events $A, C, D$.

\solution{
\begin{eqnarray*}
\prob{C \intersect A} & = & \prob{(C \intersect A) \intersect D}
    + \prob{(C \intersect A) - D},\\
& = & \prob{(C \intersect D) \intersect A}
    + \prob{(C - D) \intersect A}.\\
\end{eqnarray*}
Now dividing both sides of this equation by $\prob{A}$ yields the desired
identity.}


\problempart Calculate $\prob{H \mid B \intersect M}$, i.e., the probability that
a driver who routinely ``talks and drives" and caused a crash, caused the car crash while talking on his cell phone.  Assume that the
the probability of a driver causing a crash while NOT talking on his cell phone is not influenced by whether or not s/he routinely talks on his cell phone while driving.

\hint $\prob{C \mid A}  =  \prob{C \intersect D \mid A} + \prob{C - D \mid
A}$.

\solution{
Note that
\[
\prob{H \mid B \intersect M}
= \frac{\prob{H \intersect B \intersect M}}{\prob{B \intersect M}}
= \frac{\prob{H}}{\prob{M \mid B} \prob{B}}.
\]

To evaluate $\prob{M \mid B}$, we apply the identity from the hint:
\[
\prob{M \mid B} = \prob{M \intersect H \mid B} + \prob{M - H \mid B}.
\]
We know that probability of a driver causing a crash while NOT talking on his cell phone is not influenced by whether or not s/he routinely talks on his cell phone while driving,
namely,
\[\prob{M - H \mid B} = \prob{M - H},\]
and $H \intersect M = H$, so
\[
\prob{M \mid B} =  \prob{H \mid B} + \prob{M - H} =
\frac{1}{2000} + \frac{6K - 1.5 K}{125M}.
\]

Now we have
\begin{eqnarray*}
\prob{H \mid B \intersect M} & = & \frac{1500 / 125M}{(1/2000 +
(6K - 1.5 K)/125M)3M/125M}\\
 & = & \frac{1500}{3M/2000 + 4500 \cdot 3/125}\\
 & = & \frac{1}{1 + 9/125}\ =\ \frac{125}{134}\  \approx\  0.93.
\end{eqnarray*}
}

\problempart How could you counter Dershowitz' argument that evidence that a driver routinely talks on his cell phone while driving should be excluded from the evidence?

\solution{
In general, the probability that someone was talking on the phone during the crash is 25\%. However, if the driver routinely ``talks and drives", the probability that he was on the phone during the crash jumps to  93\%, so the fact of using a cell phone while driving greatly influences the probability of a driver's guilt.
Assuming it is proper for a jury to consider such probabilities---this is
questionable by the way---then Dershowitz' argument that talking on the phone is
unlikely to be relevant does not hold up.}

\end{problemparts}
\end{problem}
                                 