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\begin{document}

\recn{11}{4/24}{April 24, 1998}


\section{Team problems}


\begin {problems}


\problem You toss a coin twice. What is the probability that you
get two heads given that you get {\em at least one head.}
What is the probability that you  get two heads given that 
{\em the first toss} gives a head?

\problem You generate a sequence of eight numbers $n_1,\ldots,n_8$,
with $1 \leq n_i \leq  6$ for $i=1,\ldots,8$ in
the following way. You first choose at random a  position
$i \in \{ 1,2,3,4,5,6,7,8\}$ and you set $n_i =1$. Then for any $j\neq i$ you 
roll a die and you set $n_j$  according to the outcome of the die.
Observe that any sequence of 8 numbers that
contains at least one 1 can be generated in this way.
Is the distribution described above uniform, namely are all the
sequences with at least one 1 equally likely to be generated?

\bparts


\ppart Compute the probability of any given sequence by 
describing the sample space and showing which events lead
to the generation of the given sequence.
\ppart Which sequence(s) is/are most likely to be generated?

\eparts

\problem Consider the scenario of the previous section: a problem
in a quiz contains an error with probability $10 \%$, the
TA has a probability $80\%$ of answering correctly a question
of yours about whether there is an error or not; the
lecturer answers correctly your question with probability$75 \%$;
the correctness of the answer of the lecturer is independent
of the correctness of the answer of the TA.

Is the event ``the TA says that there is an error'' independent
of the event ``the lecturer says that there is an error''?

\end {problems}

\end {document}

