\documentclass[12pt]{article}\usepackage[Mickael]{ammaths}\en
\usepackage{pst-eucl}



\begin{document}\nocolor

\enteteseconde{12}{Probabilities}{Statistic and probabilities 3}



\begin{center}
\fbox{\parbox{13cm}{\sf\small A la fin de ce chapitre, vous devez \^etre capable de :
\begin{itemize}
\itemb d\'eterminer la probabilit\'e d'\'ev\'enements dans des situations d'\'equiprobabilit\'e ;
\itemb utiliser des mod\`eles d\'efinis \`a partir de fr\'equences observ\'ees ;
\itemb conna\^itre et exploiter la formule $p(A\cup B)+p(A\cap B)=p(A)+p(B)$.
\end{itemize}}}
\end{center}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exocpart{From frequencies to probabilities}

\exoc{Consider the following random experiment : roll two 6-sided dice and record the sum of the top sides.

\probpart{-- Statistics}

\begin{enumerate}
\item Can you predict with certainty the outcome of this experiment ?
\item List all the possible outcomes.
\item \begin{enumerate}
      \item Do you think that all these outcomes have the same probability to appear ?
      \item Conjecture the most probable outcome, if there is one.
      \end{enumerate}
\item \begin{enumerate}
      \item Build a \textbf{sample of size 10} for this experiment and give the absolute and relative frequencies of each sum in a table with three rows and 112 columns.
%       \begin{center}
%       \begin{tabular}{|*{12}{c|}} \hline
%       Sum     &\;2\;&\;3\;&\;4\;&\;5\;&\;6\;&\;7\;&\;8\;&\;9\;&\;10\;&\;11\;&\;12\;   \\\hline
%       Abs. Freq. & & & & & & & & &  &  &   \rule{0pt}{15pt} \\[5pt] \hline
%       Rel. Freq. & & & & & & & & &  &  &  \rule{0pt}{15pt}\\[5pt] \hline
%       \end{tabular}
%       \end{center}
      \item According to the table, what is the most probable outcome ?
      \item Did everyone in the class get the same frequency distribution ? Is that normal ?
      \end{enumerate}
\item To build larger samples, we can use a computer to simulate the experiment. Here are the results for a sample of
size $100\ 000$ :
	\begin{center}\vspace{-5pt}\hspace*{-1.5cm}
	\begin{tabular}{|*{12}{c|}} \hline
	Sum     &2   &3   &4   &5    &6    &7    &8    &    9&10  &11  &12   \\\hline
	Abs. Freq. &2893&5447&8270&11189&13809&16702&13903&11195&8348&5530&2714  \\\hline
	%Rel. Freq. & & & & & & & & &  &  &  \\\hline
	\end{tabular}
	\end{center}
\begin{enumerate}
\item Compute the relative frequency of each sum.
\item According to the table, what is the most probable outcome ?
\end{enumerate}
\end{enumerate}


\probpart{-- Probabilities}

Statistics, which are always studied after carrying out the experiment, \emph{a posteriori}, are not enough to completely understand
this experiment, as the results from one sample may be quite different from another sample. In this part, we will study
the experiment before actually carrying it out, \emph{a priori}. 

\begin{enumerate}
% \item The first row in the table below shows the possible results for the first die, and the the first column shows the
% possible results for the second die. Fill out the table with the appropriate sums.
% \begin{center}
% \begin{tabular}{|c|c|c|c|c|c|c|}
% \hline
%  &\;\;$1$\;\; &\;\; $2$\;\; &\;\; $3$\;\; &\;\; $4$\;\; &\;\; $5$\;\; &\;\; $6$\;\; \\
% \hline
% $1$ & & & & & & \rule{0pt}{15pt} \\[5pt]
% \hline
% $2$ & & & & & & \rule{0pt}{15pt} \\[5pt]
% \hline
% $3$ & & & & & & \rule{0pt}{15pt} \\[5pt]
% \hline
% $4$ & & & & & & \rule{0pt}{15pt} \\[5pt]
% \hline
% $5$ & & & & & & \rule{0pt}{15pt} \\[5pt]
% \hline
% $6$ & & & & & & \rule{0pt}{15pt} \\[5pt]
% \hline
% \end{tabular}
% \end{center}
\item Give all the possible outcomes of this experiment in a double-entry table.
\item Knowing that each cell in the table as the same probability to appear, give in a second table the probability of
each sum. Probabilities may be given as fractions or as decimal numbers to 3DP.
\item Compare the probabilities to the relative frequencies with the simulation of size $100\ 000$.
\end{enumerate}}


\exoc{In casinos, dice are changed very often to avoid wear, that could induce
irregularities. A casino has received from a new manufacturer a batch of dice, all of them truly identical, and an
employee is asked to check if these dice are well-balanced. To do so, he picks one die randomly.

\begin{enumerate}
\item First, the employee rolls the die 10 times and records the frequency of each side :
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Side & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
Frequency & $3$ & $1$ & $1$ & $2$ & $1$ & $2$ \\
\hline
\end{tabular}
\end{center}
	\begin{enumerate}
	\item Compute the relative frequency of each side and give the results in a table. Three more rows will be
added to this table in the next questions. The row with the results of the 10 throws will be labelled ``10 throws''.
	\item The employee thinks that the die is well-balanced. What do you think ?
	\end{enumerate}
\item The employee goes on by rolling the die 90 extra times, for a total of 100 throws. The results for these 100
throws are given below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Side & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
Frequency & $15$ & $11$ & $14$ & $15$ & $19$ & $26$ \\
\hline
\end{tabular}
\end{center}
	\begin{enumerate}
	\item Compute the new relative frequencies and show them in a new row in the table you did in question 1.
	\item The employee now doubts the fact that the die is well-balanced. What assumption could he logically make
at that point ?
	\end{enumerate}
\item The employee goes on by rolling the die 900 extra times, for a total of 1000 throws. The results for these 1000
throws are given below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Side & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
Frequency & $170$ & $79$ & $167$ & $176$ & $238$ & $170$ \\
\hline
\end{tabular}
\end{center}
	\begin{enumerate}
	\item Compute the new relative frequencies and show them in a new row in the table you did in question 1.
	\item Is the assumption made by the employee after the 100 throws confirmed by these results ? If not, what can
he now surmise ?
	\end{enumerate}
\item To get to the bottom of the problem, the employee enlists some of his colleagues and together they roll the die
4000 extra times, to reach a total of 5000 throws. The results are shown below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Side & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
Frequency & $855$ & $396$ & $818$ & $859$ & $1223$ & $849$ \\
\hline
\end{tabular}
\end{center}
	\begin{enumerate}
	\item Compute the new relative frequencies and show them in a new row in the table you did in question 1.
	\item Do these new values confirm the employee's inkling ? What should he tell to the casino's management ?
	\end{enumerate}
\item The probabilities of the sides of this die are in fact equal to some simple fractions. Conjecture their values.
\end{enumerate}
}

\pagebreak


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%
\exocpart{Modeling a situation}

\exoc{A card is randomly picked from a 32-cards deck, with 8 cards in four different suits : hearts ($\heartsuit$),
spades ($\spadesuit$), diamonds ($\diamondsuit$) and clubs ($\clubsuit$).
\begin{enumerate}
\item What is the probability of picking a club ?
\item What is the probability of picking a black card ?
\item What is the probability of not picking a diamond ?
\item What is the probability of picking a face card (jack, queen or king) ?
\item What is the probability of picking an ace ?
\item What is the probability of not picking a 7 ?
\item What is the probability of picking a black jack ?
\item What is the probability of picking the queen of hearts ?
\end{enumerate}
}

\exoc{In the French National Lottery (Loto), each player picks six numbers (plus one, called the complementary number,
that we won't consider in this exercise) between 1 and 49. In this exercise, we consider the random experiment that
consists in drawing a single ball from the 49 in the machine.

\begin{enumerate}
\item What is the probability of picking an even number ?
\item What is the probability of picking an odd number ?
\item What is the probability of picking a multiple of 5 ?
\item What is the probability of picking a number divisible by 7 ?
\item What is the probability of picking an even multiple of 5 ?
\item What is the probability of picking an odd multiple of 5 ?
\end{enumerate}
}






\begin{minipage}{10cm}
\exoc{The main door of a building is set up with a keyboard with three letters $A$, $B$, $C$.  The code that
activates the opening of the door is made of two letters, possibly the same.
\begin{enumerate}
\item Copy and fill the tree on the right-hand side, that gives all the possible codes :
\item How many different possible codes are there.
\item Compute the probability of the following events.
\begin{description}
\item[$T$ :] The code ends with an $A$.
\item[$I$ :] The code is made of twice the same letter.
\item[$C$ :] The code starts with an $A$.
\end{description}
\end{enumerate}}
\end{minipage}
\begin{minipage}{6cm}
 \begin{center}
	\pstree[treemode=R,nodesep=5pt]{\Tdot}
	{\pstree{\Tr{A}}{\Tr{A} \Tr{B} \Tr{C}}
	 \pstree{\Tr{B}}{\Tr{$\cdots$}\Tr{$\cdots$}\Tr{$\cdots$}}
	 \pstree{\Tr{$\cdots$}}{}
	}
	\end{center}
\end{minipage}






\exoc{Inside an urn are four balls numbered 1 to 4. A ball is randomly picked from the urn and then, without putting
back the first one, a second ball. The two numbers are multiplied to make the result of the experiment.
\begin{enumerate}
 \item Use a probability tree to model the situation.
 \item What are the possible outcomes in this random experiment ?
 \item Compute the probability distribution of this random experiment.
\end{enumerate}
}

\exoc{At a school canteen, each person's menu is made by choosing :
\begin{itemize}
\item a starter (coleslaw, pasta salad and spring rolls) ;
\item a main course (chicken with fries or roasted pork with peas) ;
\item a dessert (yogurt, ice cream or fruit).
\end{itemize}
\begin{enumerate}
\item Build a tree to count the number of possible menus.
\item How many menus include a yogurt?
\item How many menus include coleslaw and chicken ?
\item How many menus include coleslaw or chicken ?
\end{enumerate}

}

\exoc{Roger is watching an old TV set that receives only 5 programs, 3 of which are showing sport at that moment. He
chooses randomly one program to start with and then moves on to a different program. We look at the number of sport
programs he will be watching.
\begin{enumerate}
\item Use a table or a probability tree to show all the possible situations.
\item Compute the probability that Roger watches
	\begin{enumerate}
	\item no sport program ;
	\item at least one sport program ; 
	\item exactly one sport program ; 
	\item two sport programs.
	\end{enumerate}
\end{enumerate}
}


\exoc{Two fair 6-sided dice are rolled and the highest top side is recorded.
\begin{enumerate}
 \item Fill out a double-entry table to model the situation.
 \item Compute the probability distribution of this random experiment.
\end{enumerate}
}

\exoc{In a summer camp, there are 100 children. They can play two different sports, football or tennis. Of the 100
children, 60 like football, 45 like tennis and 18 like both sports.
\begin{enumerate}
\item Represent the situation with a double-entry table
\item \begin{enumerate}
      \item How many children like none of the two sports ? 
      \item How many children like only football ?
      \item How many children like only tennis ?
      \item How many children like at least one of the two sports ?
      \end{enumerate}
\end{enumerate}}



\exoc{During a campaign for road safety, the brakes and lights of 400 cars were monitored. The test showed that 60 cars
had a brake defect, 140 cars had a faulty light, and 45 cars had the two defects.
\begin{enumerate}
\item Fill out the Venn diagram below to represent the situation.
	\begin{center}
	\psset{xunit=1.5cm,yunit=0.65cm,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
	\begin{pspicture*}(-3.14,-1)(4.5,2.92)
	\rput{-0.21}(0.71,0.99){\psellipse(0,0)(3.29,1.87)}
	\rput{0.9}(-0.28,0.98){\psellipse(0,0)(1.54,0.86)}
	\rput{1.11}(1.63,0.94){\psellipse(0,0)(1.37,0.91)}
	\uput[dl](-0.5,1.5){B}
	\uput[dl](2,1.5){L}
	\end{pspicture*}
	\end{center}
\item One of the 400 cars is randomly picked. Compute the probabilities of the following events.
	\begin{enumerate}
      \item The car has a brake defect but no faulty light.
      \item The car has a faulty light but no brake defect.
      \item The car has none of these defects.
      \item The car has at least one of the two defects.
      \end{enumerate}
\end{enumerate}
}


\exoc{A group of friends are hanging out during a spring afternoon. It turns out that 60\% of them want to go to a
movie, 50\% want to go to the swimming-pool and 10\% want none of these two activities.
\begin {enumerate}
\item Illustrate the situation with a Venn diagram.
\item How many friends agree to any of these two activities ?
\item How many friends agree only to one of these two activities ?
\end {enumerate}
}

	
\begin{multicols}{2}
\exoc{A group of students are trying to organize an end-of-year party. 56 of them want to go to a pizzeria, 55 to a
Chinese restaurant and 50 to a fast-food. Furthermore, 19 refuse to go to a fast-food but agree with the two other
possibilities and 12 accept the three possibilities. Finally, 15 refuse both the pizzeria and the fast-food and 7
refuse to go to a Chinese restaurant.
\begin{enumerate}
\item Fill out the Venn diagram below.
\item What is the probability that a randomly picked student only agrees to go to a fast-food ?
\item If you were the organizer of that party, what restaurant would you choose ?
\end{enumerate}}

\columnbreak

	\begin{center}
	\psset{unit=1.3cm} \pspicture(0,-1)(6,3.5)
	%\pspolygon[linewidth=1.5pt,fillstyle=solid,fillcolor=white](0,-2)(6,-2)(6,4)(0,4)
	%\uput[dr](6,4){$\Omega$}
	\pscircle[linewidth=1.5pt,fillstyle=solid,fillcolor=white](4,2){1.5}
	\uput[u](4.2,2.5){Fast-food}
	\pscircle[linewidth=1.5pt,linecolor=gray](2,2){1.5}
	\uput[u](2,2.7){Pizzeria}
	\pscircle[linewidth=1.5pt,linestyle=dashed](3,0.5){1.5}
	\uput[u](3,-0.75){Chinese} 
% 	\uput[ur](0.5,-1.5){$1$}
% 	\uput[u](1.3,1.7){$2$} \uput[u](3,2.1){$3$} \uput[u](4.7,1.7){$4$}
% 	\uput[u](2.2,0.7){$5$} \uput[u](3,1.2){$6$} \uput[u](3.8,0.7){$7$}
% 	\uput[u](3,0){$8$}
	\endpspicture
	\end{center}
\end{multicols}



\exoc{In a dancing school, 48 students are enrolled in three different classes : break dance (25 students), jazz (18
students) and classical (22 students). Three students attend the three classes, five only do jazz and four are
enlisted in classical and jazz but not break dance.
\begin{enumerate}
 \item Build a Venn diagram to represent the situation.
 \item What is the probability that a randomly picked student is doing classical and break dance but not jazz.
\end{enumerate}}
	




\exoc{A software is used to sort messages on mail boxes.
The designers have tested 10000 e-mails and here are their conclusions:
\begin{itemize}
\item 70\% of incoming messages are spam;
\item 95\% of spam are sent to the Junk folder;
\item 2\% of good messages are sent to the Junk folder. 
\end{itemize}

\begin{enumerate}
\item Fill out the following table:
\begin{center}
 \begin{tabularx}{\linewidth}{|*{4}{>{\centering \arraybackslash}X|}}\hline
&Spam&Not spam & Total \\ \hline
Junk & & & \\ \hline
Not junk & & & \\ \hline
Total & & & 10000 \\ \hline
\end{tabularx}
\end{center}
\item We consider that this sample of 10000 is representative of all of the incoming messages, so we will use the
relative frequencies as probabilities.
\begin{enumerate}
\item Compute to 3DP the approximate values of the probabilities of the events:
\begin{enumerate}
\item a spam message is not sent to the Junk folder ;
\item a good message is sent to the Junk folder ;
\item the software is making a mistake.
\end{enumerate}
\item Compute the probability that a message sent to the Junk folder was spam.
\item Compute the probability that a message not sent to the Junk folder was good.
\end{enumerate}
\end{enumerate}
}



\exoc{A fair 4-sided die, with sides numbered 1 to 4, is thrown three times to make a three-digits number.

\begin{enumerate}
\item Build a probability tree to represent this situation.
\item List all the possible results.
\item Compute the probabilities of the following events.
\begin{description}
\item[$A$ :] The number starts with a 1.
\item[$B$ :] The number is a multiple of 3.
\item[$C$ :] The number is even.
\item[$D$ :] The number is strictly greater than 320.
\end{description}
\item A casino is offering this game to its clients, with the following rules :
\begin{itemize}
\item the player earns 10 euros if the number is made of three times the same digit ;
\item the player earns 5 euros if the same digit appears exactly twice in the number ;
\item for all other numbers, the player earns 1 euro.
\end{itemize}
	\begin{enumerate}
	\item Compute the probability of each earning.
	\item Compute the average earning of the player.
	\item The casino decides to set the price for a game to 3 euros. Will it win or lose money ?
	\end{enumerate}
\end{enumerate}
}

\exoc{An office is made of two corridors, numbered 1 and 2, and six working rooms numbered A, B, C, D, E and F.
These rooms communicate through 12 doors, as shown in the picture below :

\begin{center}
\psset{yunit=1cm}
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\psframe(9,5)
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\psline(2,1.05)(2,1.45) \psline(1.5,2.3)(1.5,2.7)\psline(1.5,3.55)(1.5,3.95)
\psline(1.5,1.05)(1.5,1.45) \psline(2,3.55)(2,3.95)
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\psline(4.25,1.05)(4.25,1.45) \psline(4.75,1.05)(4.75,1.45) 
\psline(7.25,2.3)(7.25,2.7)\psline(7.75,2.3)(7.75,2.7)
\psline(4.25,3.55)(4.25,3.95)\psline(4.75,3.55)(4.75,3.95)
\psline(7.25,3.55)(7.25,3.95)\psline(7.75,3.55)(7.75,3.95)
\psline(3,1.25)(3,1.75) \psline(2.8,1.75)(3.2,1.75)
\psline(6,1.25)(6,1.75) \psline(5.8,1.75)(6.2,1.75)
\psline(3,2.5)(3,3) \psline(2.8,3)(3.2,3)
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\psline(3,2.5)(3,2) \psline(2.8,2)(3.2,2)
\psline(6,2.5)(6,2) \psline(5.8,2)(6.2,2)
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\rput(1.75,1.8){A} \rput(4.5,1.8){B} \rput(7.5,1.8){C} 
\rput(7.5,3.1){D} \rput(4.5,3.1){E} \rput(1.75,3.1){F} 
\end{pspicture}
\end{center}

There is no door between rooms B and E.\\

\begin{itemize}
\itemb A robot, stored in corridor 1, is programmed to clean exactly three working rooms.
\itemb The robot starts its course by entering one of the rooms A, B or C.
\itemb As soon as the robot enters a room, it cleans it completely.
\itemb The robot cannot go through the same door twice, or clean the same room twice.
\itemb Once the three rooms have been cleaned, the robot returns to the corridor 1, sometimes passing through corridor 2
and a side way not shown on the picture.
\end{itemize}
\bigskip
A \emph{circuit} is any possible list of three rooms cleaned successively by the robot.
\bigskip

\textbf{Examples :}
	\begin{itemize}
	\itemb ABC and BCD are circuits.
	\itemb CBA and ABC are different circuits.
	\itemb ABE is not a possible circuit (there is no door between rooms B and E).
	\itemb DEF is not a possible circuit (the robot can't start by cleaning room D).
	\end{itemize}
\begin{enumerate}
\item Find the six possible circuits (you may use a tree to do so).\\
We admit that these six circuits have the same probability.
\item \begin{enumerate}
	\item Compute the probability $p_{1}$ of the event ``room E is the third room cleaned by the robot''.
	\item Compute the probability $p_{2}$ of the event ``the robot exits through corridor 2''.
	\end{enumerate}
\item The table below gives the time needed to clean each of the rooms (in minutes) :
\begin{center}
\begin{tabularx}{\linewidth}{|p{2.5cm}|*{6}{>{\centering \arraybackslash}X|}}\hline
Room & A&B&C&D&E&F\\ \hline
Time &20 min&24 min&30 min&14 min&22 min &14 min\\ \hline
\end{tabularx}
\end{center}

\begin{enumerate}
	\item Compute the time needed for each of the six possible circuits.
	\item Compute the average time for a circuit.
	\item Compute the probability $p_{3}$ of the event ``the robot cleans the $3$ rooms in less
than $60$~minutes''.
	\end{enumerate}
\end{enumerate}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%

\exocpart{Probabilities calculations}

\exoc{A survey has been made in early 1998 about the Internet. The answers of 1000 people are given below : 
\begin{list}{$\bullet$}{}
\item 60\% have reported not using the Internet;
\item 35\% are under 25 years old and among them 80\% have reported using the Internet;
\item 30\% are over 50 years old and among them 85\% have reported not using the Internet.
\end{list}

\begin{enumerate}
\item Fill out the following table :\medskip\\
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|c|*{2}{>{\centering}m{3.5cm}|}>{\centering}m{2.5cm}|}
\hline
& Internet & No Internet & Total \tabularnewline\hline
Under 25 years old & &  & \tabularnewline\hline
25 to 50 years old & & & \tabularnewline\hline
Over 50 years old & & & \tabularnewline\hline
Total & & & 1\,000 \tabularnewline\hline
\end{tabular}
\medskip
\item We pick randomly a person from the survey and we define the events :

$A$ : ``The person is an Internet user.''

$B$ : ``The person is under 25 years old.''

\begin{enumerate}
\item Compute the probabilities of the events $A$ and $B$.
\item Describe with a sentence the event $\bar{B}$, then compute $p(\bar{B})$.
\item Describe with a sentence the event $A\cap B$, then compute $p(A\cap B)$.
\item Describe with a sentence the event $A\cup B$, then compute $p(A\cup B)$.
\item Knowing that the person chosen is not an Internet user, what is the probability that he is under 50 years old ?
\end{enumerate}
\end{enumerate}
}


\exoc{Four tokens, marked A, B, C, D, are put in a bag. Three tokens are picked successively, without putting them back
in the bag.
\begin{enumerate}
\item Use a tree to describe all the possible outcomes.
\item We consider the following events :
      \begin{itemize}
      \itemb E : ``The first token is marked with the letter B.''
      \itemb F : ``The token C hasn't been picked.''
      \end{itemize}
      Compute the probabilities of the events E and F.
\item\begin{enumerate}
      \item What outcomes make up the event $E \cap F$ ?
      \item Compute the probability of the event $E \cap F$. 
      \end{enumerate}
\item Deduce the probability of the event $E \cup F$. 
\end{enumerate} 
}



\exoc{A random experiment is defined as follows : throw a fair six-sided die, then a coin and count 1 for heads and 2
for tails. Multiply the number from the die by the number from the coin. 
\begin{enumerate}
\item Represent the experiment with a probability tree.
\item At the end of each branch of the tree, show the outcome of the experiment.
\item In a table, show the probability distribution of the experiment.
\item Compute the probabilities of the following events :
	\begin{description}
	\item[$A$ :] the result is even  ;
	\item[$B$ :] the result is a multiple of 3 ;
	\item[$C$ :] the result is neither 6, nor 5 ;
	\item[$D$ :] the result is greater than or equal to 4 ;
	\item[$E$ :] the result is strictly less than 6. 
	\end{description}
\item \begin{enumerate}
	\item List all the outcomes making up the event $A\cap B$.
	\item Deduce the probability of the event $A\cap B$.
	\end{enumerate}
\item \begin{enumerate}
	\item List all the outcomes making up the event $A\cup B$.
	\item Deduce the probability of the event $A\cup B$.
	\item Is it true that $P(A\cup B)=P(A)+P(B)$ ? If so, prove it. If not, correct the formula.
	\end{enumerate}
\end{enumerate}
}

\pagebreak

\exoc{A six-sided fair die is rolled. Let's consider the following events:
\begin{itemize}
 \itemb $A$ : ``You get the number 1.''
 \itemb $B$ : ``You get a number strictly greater than 2.''
 \itemb $C$ : ``You get an even number.''
\end{itemize}

\begin{enumerate}
\item List all the outcomes making up the events $A$, $B$ and $C$. 
\item Represent the set of all possible outcomes $E$ and the sets $A$, $B$ and $C$ with a diagram.
\item List all the outcomes making up the events $A\cup B$ and $A\cap B$.
\item Find the probabilities of the events $A$, $B$, $C$, $A\cup B$ and $A\cap B$.
\end{enumerate}
}

\exoc{An industrial production line consists of two machines called M1 and M2 in this exercise.
Surveys have shown that
\begin{itemize}
\itemb the probability of the event $A$ ``the machine M1 is operational'' is $96,7$ \% ;
\itemb the probability of the event $B$ ``the machine M2 is operational'' is $98,2$ \%.
\end{itemize}
\begin{enumerate}
\item The probability that the two machines are operational at the same time is $95,5$ \%. Translate this sentence into
mathematical notation.
\item Translate the following events into English, then find their probabilities.
$$A, B, \overline{A}, \overline{B}, A\cup B, \overline{A\cup B}.$$
\end{enumerate}
}



\exoc{A car salesman has noticed that 70\% of his clients buy a brand new car, of which 80\% choose a French brand. All
in all, the French brands account for three quarters of his sales. Let's consider a random client and define the
following events :
\begin{itemize}
\itemb $N$ : ``the client buys a brand new car'' ;
\itemb $F$ : ``the client buys a French car''.
\end{itemize}

\begin{enumerate}
\item Represent the information in the text with a table or a diagram.
\item Translate each of the following events into English and compute their probabilities.
$$N, F, \overline{N}, \overline{F}, N\cap F, \overline{N\cap F}, N\cup F, \overline{N\cup F}.$$
\end{enumerate}
}



\exoc{A company produces pocket calculators in large quantity. The calculators can have two types of flaws.
: the A flaw or the B flaw. After a statistical study on 800 calculators it turns out that :
\begin{itemize}
\itemb  15\% of the calculators have the A flaw ; 
\itemb  among the calculators having the A flaw, 95 \% do not have the B flaw ; 
\itemb  among the calculators that do not have the A flaw, 5 \% have the B flaw.
\end{itemize}
\begin{enumerate}
\item Copy and fill out the following double-entry table :
      \begin{center}
      \begin{tabular}{|l|*{2}{p{3cm}|}c|}\hline
      \rule[-3mm]{0cm}{8mm} & A flaw & Not the A flaw &Total  \\\hline
      \rule[-3mm]{0cm}{8mm} B flaw      &         &               &          \\\hline
      \rule[-3mm]{0cm}{8mm} Not the B flaw&         & 				 &          \\\hline
      \rule[-3mm]{0cm}{8mm}Total & & &800 \\\hline
      \end{tabular}
      \end{center}
\item One calculator is randomly picked. 
      \begin{enumerate}
      \item Give the probability of the event $A$ : ``The calculator has the A flaw''.
      \item Give the probability of the event $B$ : ``The calculator has the B flaw''.
      \item Define the events $A\cap B$ and $A\cup B$, then compute their probabilities.
      \item What is the probability of the event : ``The calculator has no flaw''.
      \end{enumerate}
\item A calculator with the A flaw has been chosen. What is the probabilty that it also has the B flaw ?
\end{enumerate}
}




\exoc{In a class, there are 24 students, including 9 boys. Among the boys, 7 are half-boarders, and none are boarders.
In the whole class, there are twice less day students then boarders and half-boarders all together, and there is only
one girl which is a full-boarder.

A student from that class is randomly picked, and we define the following events :
\begin{multicols}{2}
\begin{description}
\item[G :] the student is a girl ;
\item[B :] the student is a boy ;
\item[D :] the student is a day student ;
\item[H :] the student is a half-boarder ;
\item[F :] the student is a boarder.
\end{description}
\ 

\end{multicols}

\begin{enumerate}
\item Sum up the information from the text in a double-entry table.
\item Translate into English the following events
	\begin{multicols}{4}
	\begin{enumerate}
	\item $F\cup H$ ;
	\item $H\cup D$ ;
	\item $G\cap D$ ;
	\item $B\cap H$ ;
	\item $G\cap B$ ;
	\item $\bar{G}$ ;
	\item $\bar{F}$.
	\end{enumerate}
	\end{multicols}
\item Use the table to compute the probabilities of these events.
\end{enumerate}
}

\exoc{\tice\ A coin is thrown repeatedly, a certain number of times.
\begin{enumerate}
\item If the coin is thrown just once, what is the probability of getting a head ?
\item If the coin is thrown 2 times, what is the probability of getting only heads ?
\item If the coin is thrown 3 times, what is the probability of getting only heads ?
\item If the coin is thrown $n$ times, with $n$ a natural number, what is the probability of getting only heads ?
\item How many times must the coin be thrown so that the probability of getting only heads is less than 0.001 ?
\item Consider the algorithm below :
\begin{center}
\begin{minipage}[t]{12cm}
\IncMargin{1em}
\begin{algorithm}[H]
\Begin{\KwIn{$e$, a real number \;}
$1\rightarrow n$ \;
$0.5\rightarrow p$ \;
\While{$p\geq e$}{$n+1\rightarrow n$ \;
	$0.5^n\rightarrow p$ \;}
\KwOut{$n$ \;}
}
\end{algorithm}
\DecMargin{1em}
\end{minipage}
\end{center}
	\begin{enumerate}
	\item Apply the algorithm with $e=0.1$, with $e=0.01$ and with $e=0.001$.
	\item What is the aim of this algoritm ?
	\item How would you modify the algorithm for a biaised coin, where the probability of getting a head is entered by the user ?
	\end{enumerate}


\end{enumerate}


}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\en
\partie{Homework \#8 -- The minimal depth}

Roger, the farmer, noticed that his cows drink at least 2000 liters per day. Below is the shape of the trough that he want to built. The bottom $DD'CC'$ is a 1x2 m. rectangle and the front face $ABCD$ is a isosceles trapezium. What will be the minimal depth (denoted $h$ on the plan) of the trough ?

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\end{document}
