bac-s-maths 2025 Q2

bac-s-maths · France · bac-spe-maths__asie_j1 Conditional Probability Sequential/Multi-Stage Conditional Probability

A company that manufactures toys must perform conformity checks before their commercialization. In this exercise, we are interested in two tests performed by the toy company: a manufacturing test and a safety test. Following a large number of verifications, the company claims that:

$95 \%$ of toys pass the manufacturing test;
Among toys that pass the manufacturing test, $98 \%$ pass the safety test;
$1 \%$ of toys pass neither of the two tests.

A toy is chosen at random from the toys produced. We denote:

F the event: ``the toy passes the manufacturing test'';
S the event: ``the toy passes the safety test''.

Part A

From the data in the statement, give the probabilities $P ( F )$ and $P _ { F } ( S )$.
a. Construct a probability tree that illustrates the situation with the data available in the statement. b. Show that $P _ { \bar { F } } ( \bar { S } ) = 0.2$.
Calculate the probability that the chosen toy passes both tests.
Show that the probability that the toy passes the safety test is 0.97 rounded to the nearest hundredth.
When the toy has passed the safety test, what is the probability that it passes the manufacturing test? Give an approximate value of the result to the nearest hundredth.

Part B
A batch of $n$ toys is randomly selected from the company's production, where $n$ is a strictly positive integer. We assume that this selection is made from a sufficiently large quantity of toys to be assimilated to a succession of $n$ independent draws with replacement. Recall that the probability that a toy passes the manufacturing test is equal to 0.95. Let $S _ { n }$ be the random variable that counts the number of toys that have passed the manufacturing test. We admit that $S _ { n }$ follows the binomial distribution with parameters $n$ and $p = 0.95$.

Express the expectation and variance of the random variable $S _ { n }$ as a function of $n$.
In this question, we set $n = 150$. a. Determine an approximate value to $10 ^ { - 3 }$ of $P \left( S _ { 150 } = 145 \right)$. Interpret this result in the context of the exercise. b. Determine the probability that at least $94 \%$ of the toys in this batch pass the manufacturing test. Give an approximate value of the result to $10 ^ { - 3 }$.
In this question, the non-zero natural integer $n$ is no longer fixed.

Let $F _ { n }$ be the random variable defined by: $F _ { n } = \frac { S _ { n } } { n }$. The random variable $F _ { n }$ represents the proportion of toys that pass the manufacturing test in a batch of $n$ toys selected. We denote $E \left( F _ { n } \right)$ the expectation and $V \left( F _ { n } \right)$ the variance of the random variable $F _ { n }$. a. Show that $E \left( F _ { n } \right) = 0.95$ and that $V \left( F _ { n } \right) = \frac { 0.0475 } { n }$. b. We are interested in the following event $I$: ``the proportion of toys that pass the manufacturing test in a batch of $n$ toys is strictly between $93 \%$ and $97 \%$''. Using the Bienaymé-Chebyshev inequality, determine a value $n$ of the size of the batch of toys to select, from which the probability of event $I$ is greater than or equal to 0.96.

A company that manufactures toys must perform conformity checks before their commercialization. In this exercise, we are interested in two tests performed by the toy company: a manufacturing test and a safety test.\\
Following a large number of verifications, the company claims that:
\begin{itemize}
  \item $95 \%$ of toys pass the manufacturing test;
  \item Among toys that pass the manufacturing test, $98 \%$ pass the safety test;
  \item $1 \%$ of toys pass neither of the two tests.
\end{itemize}
A toy is chosen at random from the toys produced. We denote:
\begin{itemize}
  \item F the event: ``the toy passes the manufacturing test'';
  \item S the event: ``the toy passes the safety test''.
\end{itemize}

\textbf{Part A}
\begin{enumerate}
  \item From the data in the statement, give the probabilities $P ( F )$ and $P _ { F } ( S )$.
  \item a. Construct a probability tree that illustrates the situation with the data available in the statement.\\
b. Show that $P _ { \bar { F } } ( \bar { S } ) = 0.2$.
  \item Calculate the probability that the chosen toy passes both tests.
  \item Show that the probability that the toy passes the safety test is 0.97 rounded to the nearest hundredth.
  \item When the toy has passed the safety test, what is the probability that it passes the manufacturing test? Give an approximate value of the result to the nearest hundredth.
\end{enumerate}

\textbf{Part B}\\
A batch of $n$ toys is randomly selected from the company's production, where $n$ is a strictly positive integer. We assume that this selection is made from a sufficiently large quantity of toys to be assimilated to a succession of $n$ independent draws with replacement.\\
Recall that the probability that a toy passes the manufacturing test is equal to 0.95. Let $S _ { n }$ be the random variable that counts the number of toys that have passed the manufacturing test. We admit that $S _ { n }$ follows the binomial distribution with parameters $n$ and $p = 0.95$.
\begin{enumerate}
  \item Express the expectation and variance of the random variable $S _ { n }$ as a function of $n$.
  \item In this question, we set $n = 150$.\\
a. Determine an approximate value to $10 ^ { - 3 }$ of $P \left( S _ { 150 } = 145 \right)$. Interpret this result in the context of the exercise.\\
b. Determine the probability that at least $94 \%$ of the toys in this batch pass the manufacturing test. Give an approximate value of the result to $10 ^ { - 3 }$.
  \item In this question, the non-zero natural integer $n$ is no longer fixed.
\end{enumerate}
Let $F _ { n }$ be the random variable defined by: $F _ { n } = \frac { S _ { n } } { n }$. The random variable $F _ { n }$ represents the proportion of toys that pass the manufacturing test in a batch of $n$ toys selected.\\
We denote $E \left( F _ { n } \right)$ the expectation and $V \left( F _ { n } \right)$ the variance of the random variable $F _ { n }$.\\
a. Show that $E \left( F _ { n } \right) = 0.95$ and that $V \left( F _ { n } \right) = \frac { 0.0475 } { n }$.\\
b. We are interested in the following event $I$: ``the proportion of toys that pass the manufacturing test in a batch of $n$ toys is strictly between $93 \%$ and $97 \%$''.\\
Using the Bienaymé-Chebyshev inequality, determine a value $n$ of the size of the batch of toys to select, from which the probability of event $I$ is greater than or equal to 0.96.

Paper Questions

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