bac-s-maths 2016 Q3

bac-s-maths · France · polynesie 5 marks Exponential Distribution
Exercise 3
Part A
An astronomer responsible for an astronomy club observed the sky one August evening in 2015 to see shooting stars. He made observations of the waiting time between two appearances of shooting stars. He then modelled this waiting time, expressed in minutes, by a random variable $T$ which follows an exponential distribution with parameter $\lambda$. By exploiting the data obtained, he established that $\lambda = 0,2$.
  1. When the group sees a shooting star, verify that the probability that it waits less than 3 minutes to see the next shooting star is approximately 0,451.
  2. When the group sees a shooting star, what minimum duration must it wait to see the next one with a probability greater than 0,95? Round this time to the nearest minute.
  3. The astronomer has planned an outing of two hours. Estimate the average number of observations of shooting stars during this outing.

Part B
This manager sends a questionnaire to his members to get to know them better. He obtains the following information:
  • $64 \%$ of the people surveyed are new members;
  • $27 \%$ of the people surveyed are former members who own a personal telescope;
  • $65 \%$ of new members do not have a personal telescope.

  1. A member is chosen at random. Show that the probability that this member owns a personal telescope is 0,494.
  2. A member is chosen at random from among those who own a personal telescope. What is the probability that this is a new member? Round to $10 ^ { - 3 }$ near.

Part C
For practical reasons, the astronomer responsible for the club would like to install an observation site on the heights of a small town of 2500 inhabitants. But light pollution due to public lighting harms the quality of observations. To try to convince the town hall to cut off the night lighting during observation nights, the astronomer conducts a random survey of 100 inhabitants and obtains 54 favourable opinions on cutting off the night lighting. The astronomer makes the hypothesis that $50 \%$ of the village population is in favour of cutting off the night lighting. Does the result of this survey lead him to change his mind? 
\section*{Exercise 3}
\section*{Part A}
An astronomer responsible for an astronomy club observed the sky one August evening in 2015 to see shooting stars. He made observations of the waiting time between two appearances of shooting stars. He then modelled this waiting time, expressed in minutes, by a random variable $T$ which follows an exponential distribution with parameter $\lambda$. By exploiting the data obtained, he established that $\lambda = 0,2$.

\begin{enumerate}
  \item When the group sees a shooting star, verify that the probability that it waits less than 3 minutes to see the next shooting star is approximately 0,451.
  \item When the group sees a shooting star, what minimum duration must it wait to see the next one with a probability greater than 0,95? Round this time to the nearest minute.
  \item The astronomer has planned an outing of two hours. Estimate the average number of observations of shooting stars during this outing.
\end{enumerate}

\section*{Part B}
This manager sends a questionnaire to his members to get to know them better. He obtains the following information:
\begin{itemize}
  \item $64 \%$ of the people surveyed are new members;
  \item $27 \%$ of the people surveyed are former members who own a personal telescope;
  \item $65 \%$ of new members do not have a personal telescope.
\end{itemize}

\begin{enumerate}
  \item A member is chosen at random. Show that the probability that this member owns a personal telescope is 0,494.
  \item A member is chosen at random from among those who own a personal telescope. What is the probability that this is a new member? Round to $10 ^ { - 3 }$ near.
\end{enumerate}

\section*{Part C}
For practical reasons, the astronomer responsible for the club would like to install an observation site on the heights of a small town of 2500 inhabitants. But light pollution due to public lighting harms the quality of observations. To try to convince the town hall to cut off the night lighting during observation nights, the astronomer conducts a random survey of 100 inhabitants and obtains 54 favourable opinions on cutting off the night lighting.\\
The astronomer makes the hypothesis that $50 \%$ of the village population is in favour of cutting off the night lighting. Does the result of this survey lead him to change his mind?
Paper Questions
Q1 Q2 Q3 Q4a Q4b