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\begin{document}

\recn{11}{11/20}{November 20, 1998}

\section{Team Problem}

\begin{problems}

\problem 
Consider the scenario considered earlier:
a quiz problem has an error with probability $10 \%$, 
the TA says whether or not there is an error, answering correctly 
with probability $80\%$,
and the lecturer answers correctly with probability $75 \%$.
The three events of existence of an error, the TA responding correctly
and the lecturer responding correctly are mutually independent.

Is the event $T$, ``the TA says that there is an error'', independent
of the event $L$, ``the lecturer says that there is an error''?

\problem 
This is a real problem that came from a research project in
computational biology last year.  

It is known that atoms occur in different isotopes. 
For example, Carbon atoms usually have 6 protons and 
6 neutrons, but occasionally they have 7 neutrons. 
In fact, $1.11\%$ of all Carbon atoms have 7 neutrons
and $98.89\%$ have 6. From what we know, the appearance
of the different isotopes seems to be a random event, unrelated
to the functionality of the atom. 
We can tell which Carbon
atoms have an extra neutron only by their atomic weight.
The atomic weight of the generic Carbon atom is 12 while the weight of
its rarer isotope is 13.

The research team is studying proteins composed of Carbon, Oxygen,
Nitrogen and Hydrogen (the four building blocks of life).
Their atomic weights and isotopes 
are listed below. The second column is the atomic weight and 
the third column is the frequency of the isotope. Assume that the
probability of an unlisted isotope ocurring is zero (i.e., no tritium in
our model).

\begin{eqnarray*}
Hydrogen & 1 & P_{H,1}\\
 & 2 & P_{H,2}\\
& & \\
Carbon & 12 & P_{C,12}\\
 & 13 & P_{C,13}\\
& &\\
Nitrogen & 14 & P_{N,14}\\
& 15 & P_{N,15}\\
& & \\
Oxygen & 15 & P_{O,15}\\
& 16 & P_{O,16}\\
& 17 & P_{O,17}
\end{eqnarray*}

Suppose we want to predict the distribution of atomic weights for any
protein. Describe a simple algorithm for computing the weight
distribution of any protein that contains $n_C$ carbon atoms, $n_O$
oxygen atoms, $n_N$ nitrogen atoms and $n_H$ hydrogen atoms. 
[Hint: Use generating functions!]

\end {problems}


\end{document}
