According to the health authorities of a country, $7\%$ of the inhabitants are affected by a certain disease. In this country, a test is developed to detect this disease. This test has the following characteristics:
For sick individuals, the test gives a negative result in $20\%$ of cases;
For healthy individuals, the test gives a positive result in $1\%$ of cases.
A person is chosen at random from the population and tested. Consider the following events:
$M$ ``the person is sick'';
$T$ ``the test is positive''.
Calculate the probability of the event $M \cap T$. You may use a probability tree.
Prove that the probability that the test of the randomly chosen person is positive is $0.0653$.
In the context of disease screening, is it more relevant to know $P_M(T)$ or $P_T(M)$?
In this question, consider that the randomly chosen person had a positive test. What is the probability that they are sick? Round the result to $10^{-2}$ near.
People are chosen at random from the population. The size of the population of this country allows us to treat this sampling as drawing with replacement. Let $X$ be the random variable that gives the number of individuals with a positive test among 10 people. a. Specify the nature and parameters of the probability distribution followed by $X$. b. Determine the probability that exactly two people have a positive test. Round the result to $10^{-2}$ near.
Determine the minimum number of people to test in this country so that the probability that at least one of them has a positive test is greater than $99\%$.
According to the health authorities of a country, $7\%$ of the inhabitants are affected by a certain disease.\\
In this country, a test is developed to detect this disease. This test has the following characteristics:
\begin{itemize}
\item For sick individuals, the test gives a negative result in $20\%$ of cases;
\item For healthy individuals, the test gives a positive result in $1\%$ of cases.
\end{itemize}
A person is chosen at random from the population and tested.\\
Consider the following events:
\begin{itemize}
\item $M$ ``the person is sick'';
\item $T$ ``the test is positive''.
\end{itemize}
\begin{enumerate}
\item Calculate the probability of the event $M \cap T$. You may use a probability tree.
\item Prove that the probability that the test of the randomly chosen person is positive is $0.0653$.
\item In the context of disease screening, is it more relevant to know $P_M(T)$ or $P_T(M)$?
\item In this question, consider that the randomly chosen person had a positive test.\\
What is the probability that they are sick? Round the result to $10^{-2}$ near.
\item People are chosen at random from the population. The size of the population of this country allows us to treat this sampling as drawing with replacement.\\
Let $X$ be the random variable that gives the number of individuals with a positive test among 10 people.\\
a. Specify the nature and parameters of the probability distribution followed by $X$.\\
b. Determine the probability that exactly two people have a positive test. Round the result to $10^{-2}$ near.
\item Determine the minimum number of people to test in this country so that the probability that at least one of them has a positive test is greater than $99\%$.
\end{enumerate}