Every day he works, Paul must go to the station to reach his workplace by train. To do this, he takes his bicycle two times out of three and, if he does not take his bicycle, he takes his car.
When he takes his bicycle to reach the station, Paul misses the train only once in 50, whereas when he takes his car to reach the station, Paul misses his train once in 10. We consider a random day on which Paul will be at the station to catch the train that will take him to work. We denote:
V the event ``Paul takes his bicycle to reach the station'';
R the event ``Paul misses his train''. a. Draw a weighted tree summarizing the situation. b. Show that the probability that Paul misses his train is equal to $\frac { 7 } { 150 }$. c. Paul has missed his train. Determine the exact value of the probability that he took his bicycle to reach the station.
A random month is chosen during which Paul went to the station 20 days to reach his workplace according to the procedures described in the preamble. We assume that, for each of these 20 days, the choice between bicycle and car is independent of the choices on other days. We denote $X$ the random variable giving the number of days Paul takes his bicycle over these 20 days. a. Determine the distribution followed by the random variable $X$. Specify its parameters. b. What is the probability that Paul takes his bicycle exactly 10 days out of these 20 days to reach the station? Round the probability sought to $10 ^ { - 3 }$. c. What is the probability that Paul takes his bicycle at least 10 days out of these 20 days to reach the station? Round the probability sought to $10 ^ { - 3 }$. d. On average, how many days over a randomly chosen period of 20 days to reach the station does Paul take his bicycle? Round the answer to the nearest integer.
In the case where Paul goes to the station by car, we denote $T$ the random variable giving the travel time needed to reach the station. The duration of the journey is given in minutes, rounded to the minute. The probability distribution of $T$ is given by the table below:
$k$ (in minutes)
10
11
12
13
14
15
16
17
18
$P ( T = k )$
0,14
0,13
0,13
0,12
0,12
0,11
0,10
0,08
0,07
Determine the expected value of the random variable $T$ and interpret this value in the context of the exercise.
Every day he works, Paul must go to the station to reach his workplace by train. To do this, he takes his bicycle two times out of three and, if he does not take his bicycle, he takes his car.
\begin{enumerate}
\item When he takes his bicycle to reach the station, Paul misses the train only once in 50, whereas when he takes his car to reach the station, Paul misses his train once in 10.\\
We consider a random day on which Paul will be at the station to catch the train that will take him to work.\\
We denote:
\begin{itemize}
\item V the event ``Paul takes his bicycle to reach the station'';
\item R the event ``Paul misses his train''.\\
a. Draw a weighted tree summarizing the situation.\\
b. Show that the probability that Paul misses his train is equal to $\frac { 7 } { 150 }$.\\
c. Paul has missed his train. Determine the exact value of the probability that he took his bicycle to reach the station.
\end{itemize}
\item A random month is chosen during which Paul went to the station 20 days to reach his workplace according to the procedures described in the preamble.\\
We assume that, for each of these 20 days, the choice between bicycle and car is independent of the choices on other days.\\
We denote $X$ the random variable giving the number of days Paul takes his bicycle over these 20 days.\\
a. Determine the distribution followed by the random variable $X$. Specify its parameters.\\
b. What is the probability that Paul takes his bicycle exactly 10 days out of these 20 days to reach the station? Round the probability sought to $10 ^ { - 3 }$.\\
c. What is the probability that Paul takes his bicycle at least 10 days out of these 20 days to reach the station? Round the probability sought to $10 ^ { - 3 }$.\\
d. On average, how many days over a randomly chosen period of 20 days to reach the station does Paul take his bicycle? Round the answer to the nearest integer.
\item In the case where Paul goes to the station by car, we denote $T$ the random variable giving the travel time needed to reach the station. The duration of the journey is given in minutes, rounded to the minute. The probability distribution of $T$ is given by the table below:
\end{enumerate}
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | }
\hline
$k$ (in minutes) & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\
\hline
$P ( T = k )$ & 0,14 & 0,13 & 0,13 & 0,12 & 0,12 & 0,11 & 0,10 & 0,08 & 0,07 \\
\hline
\end{tabular}
\end{center}
Determine the expected value of the random variable $T$ and interpret this value in the context of the exercise.