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\begin{document}
\pset{9}{April 14, 1998}{Tuesday, April 28}

\reading Rosen \S4.4--5; 6.042 Fall 97, Notes for Lectures 18-20

\begin{problems}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 1
\problem [15 pts]

Suppose the game {\em Let's Make A Deal} is changed slightly. Instead of having 3 doors with 1 grand prize, in the new game there are 4 doors with 2 grand prizes. The rule of the game remains the same. Therefore the game now becomes:

\begin{quote}
At one stage of the game, a contestant is shown 4 doors. The contestant knows that there are 2 grand prizes behind 2 of the doors and that there are goats behind the 2 other doors. At the beginning, the contestant picks a door. To build suspense, the assistant of the game then opens a {\em different} door, revealing a goat. The contestant can then stick with his original door or switch to another unopened door. He wins a prize only if he now picks the door with the prize.
\end{quote}

In the new game, what is the probability of winning with the ``stick'' strategy? How about the ``switch'' strategy? Can you explain why the two probabilities do not sum up to 1?   

%Suppose the game {\em Let's Make A Deal} is changed slightly. Instead of having 3 doors with 1 grand prize, in the new game there are 4 doors with 2 grand prizes. The rule of the game remains the same. The player has 1 chance to switch after a door without prize is opened. What is the probability of winning with the ``stick'' strategy? How about the ``switch'' strategy? Can you explain why the two probabilities do not sum up to 1?   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 2

\problem [20 pts]

A sequence of numbers $<{\tt 0},{\tt 2},{\tt 2},{\tt 2},{\tt 2},{\tt 5},{\tt 5},{\tt 7},{\tt 7}>$ is randomly permuted to form a 
9-digit integer, in such a way that the digit {\tt 0} does not come first. (In other words, if 
the digit {\tt 0} comes first in the permutation, we try a new permutation until this 
does not occur.) Let $p$ be the 9-digit number. Find the probabilities of the
following:

\bparts
\ppart $p$ is a multiple of 2.
%\ppart  $p$ is a multiple of 3.
\ppart  $p$ is a multiple of 25.
\ppart  the first and the last digits of $p$ are {\tt 2}.
\ppart $p$ is symmetric. (i.e., the number does not change if the sequence of digits is reversed)
\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 3

\problem [10 pts]

In a lottery, 6 distinct integers are drawn randomly from 1 to 45 without replacement. What is the 
probability that the maximum of the 6 numbers is $k$, where $1\leq k\leq45$?


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Problem 4 (Modified from Fall 96 Pset 10 Problem 3)
\problem 

In a complex system with 1,000 different components, each component is
known to fail with probability at most $\frac{1}{10,000}$. The failures may not be independent events. 

\bparts
\ppart Prove that $\frac{1}{10}$ is an upper bound for the probability that one or more components in the system fail. [10 pts]

\ppart Show that this upper bound is optimal by giving a case where the probability that one or more components in the system fail is exactly $\frac{1}{10}$. [5 pts]
\eparts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 5
\problem 
There are 4 coins in a box, namely $c_1, c_2, c_3, c_4$. The probability $p_i$ that the coin $c_i$ gives a Head is $\frac{1}{i}$ for $1\leq i\leq4$. A coin is selected from the box randomly, and is being tossed until a Head is seen. 

\bparts
\ppart
Write down the sample space of the experiment. [2 pts]
 
\ppart
What is the probability that a Head is first seen in the $2^{nd}$ toss? [9 pts]

\ppart
Given that a Head is first seen in the $2^{nd}$ toss, what is the 
probability that $c_i$ is selected from the box? [9 pts]

\eparts


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 6 (Left for Problem Set 10)
\problem [10 pts]

$A$, $B$, $C$ are events of an experiment. If $C$ is independent of $A$ and $C$ is independent of $B$, is $C$ independent of $A\cap B$? Prove your answer or give a counterexample. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 7
\problem [10 pts]

A house is installed with a fire alarm to prevent fire accidents. If there 
is a fire, the probability for the fire alarm being turned on is 96\%. 
Even without a fire, the alarm may be turned on with a probability of 2\%. 
Suppose the probability for a fire accident to happen in the house is 1\%. 
If the fire alarm is now on, what is the conditional probability that there is really a fire accident?


\end{problems}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Alternative Problem 4
%\problem 
%
%In another lucky draw, a box contains 3 gold balls and 97 black balls.
%100 participants draw a ball randomly from the box in order without replacement. Only the 3 players who have drawn the gold ball get a prize. What is the probability that:  
%\bparts 
%\ppart the first player gets a prize? [2 pts]
%\ppart the last player gets a prize? [3 pts]
%\ppart the $30^{th}$ player gets a prize? [5 pts]
%\eparts











