%%%%%%%%%%%%%%  Fall 2000  PS 11.5   %%%%%%%%%%%%%%   %%%%%%%%%%%%%%%
%[[[expectation, variance]]]
%Spring 00, problem set 10.5, problem 5
\begin{problem}
Psychological diagnosis of sociopaths is notoriously unreliable.
For example, a random psychologist is as likely to diagnose a normal
patient as sociopathic as he is to recognize that the patient is normal.
A random psychologist is actually slightly \emph{less} likely to correctly
diagnose a real sociopath than not---sociopaths are really good liars.

\begin{problemparts}

\problempart In these circumstances it might seem there was no reliable way to
diagnose sociopaths, but in theory there is a way---if we have a large
enough population of psychologists who reach their judgements
independently.  Explain how to obtain a reliable diagnosis in this case.
Briefly explain why your method will be reliable, citing appropriate
results from class.  A qualitative answer is all that's required---you
need not come up with any numerical estimates or growth rates.

\solution{

Get a reliable test by repeatedly asking randomly chosen psychologists to
do the diagnosis and computing the fraction, $f$, of ``normal'' diagnoses.
Let $s>1/2$ be the probability of a ``normal'' diagnosis on a sociopathic
patient.  Diagnose normal if $f$ is closer to $1/2$ than to $s$.

By diagnosing independently enough times, there is small probability that
$f$ differs from the expected value of a diagnosis by very much (the
expected value is either 1/2 or $s$ depending on whether or not the
patient is normal---we don't know which).  By the Law of Large Numbers, by
Chebyshev's bound, or by the estimation of the binomial distribution as
used in the Gallup Poll example in Lecture Notes, if we diagnose enough
times, we can make this probability as small as desired, so the
probability of a correct diagnosis can be as close to 1 as desired.

}

\problempart Is it plausible to assume that psychologists will make their
diagnoses independently?  Briefly explain.

\solution{

Not really.  It figures that some sociopaths would be particularly good
liars and be able to fool most psychologists.  Conversely, there figure to
be normal people whose anxious or defensive manner makes a sociopathic
diagnosis by many psychologists more likely.  Also, there may be some
specially expert psychologists whose diagnoses are more reliable than
average; this implies if one expert diagnoses ``normal,'' the diagnosis is
likely to be correct, making it more likely that another expert will also
correctly diagnose ``normal,'' so their diagnoses will not be independent
of each other.
}
\end{problemparts}
\end{problem}