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\title{General Mathematics and Computational Science I}
\author{Exercise 13}
\date{October 25, 2007}
\maketitle

\begin{enumerate}

\item It is believed that 2 genes $A$ and $B$ may play some part in
the susceptibility of an individual to a disease. Of 100 patients
investigated, 17 carry gene $A$, 33 carry gene $B$ and 67 carry
neither. Find the probability that a patient carries only gene $A$,
only gene $B$ or both.

\item Prove the \emph{arithmetic-geometric-mean inequality} for $n=3$.

Note: This is Problem 9 from Ivanov, p.\ 48, which contains a sketch
of a proof.  The task here is to write out a complete self-contained
solution without reference to Ivanov.

\item Show that
\begin{equation*}
  a^2+b^2+c^2 \geq ab+bc+ca
\end{equation*}
for arbitrary real numbers $a,b$ and $c$. When does equality holds?

\end{enumerate}
\end{document}

\item Suppose you meet $n$ random people (excluding yourself).

\begin{enumerate}
\item What is the probability that at least one of them is born on the
same day of the year as you?
\item How large should $n$ be so that this happens with a probability
of at least $\tfrac12$?
\end{enumerate}

\item Show that
\begin{equation*}
  \sqrt{a + b^4} \leq \sqrt{a} + b^2 \leq \sqrt{2(a+b^4)}
\end{equation*}
for arbitrary non-negative real numbers $a$ and $b$. When does
equality holds?

\textbf{Hint:} For every real number $a\geq 0$ there exists a real
number $t$ such that $a=t^2$.