Paratuberculosis is an infectious digestive disease that affects cows. It is caused by the presence of a bacterium in the cow's intestine. A study is conducted in a region where $0.4\%$ of the cow population is infected. There is a test that reveals the immune response of an organism infected by the bacterium. The result of this test can be either ``positive'' or ``negative''. A cow is chosen at random from the region. Given the characteristics of the test, we know that:
If the cow is affected by the infection, the probability that its test is positive is 0.992;
If the cow is not affected by the infection, the probability that its test is negative is 0.984.
We denote by $I$ the event ``the cow is affected by the infection'' and $T$ the event ``the cow presents a positive test''. We denote by $\bar{I}$ the complementary event of $I$ and $\bar{T}$ the complementary event of $T$. Part A
Reproduce and complete the weighted tree below modelling the situation.
a. Calculate the probability that the cow is not affected by the infection and that its test is negative. Give the result to $10^{-3}$ near. b. Show that the probability, to $10^{-3}$ near, that the cow presents a positive test is approximately equal to 0.020. c. The ``positive predictive value of the test'' is the probability that the cow is affected by the infection given that its test is positive. Calculate the positive predictive value of this test. Give the result to $10^{-3}$ near. d. The test gives incorrect information about the cow's state of health when the cow is not infected and presents a positive test result or when the cow is infected and presents a negative test result. Calculate the probability that this test gives incorrect information about the cow's state of health. Give a result to $10^{-3}$ near.
Part B
When a sample of 100 cows is chosen at random from the region, this choice is treated as a draw with replacement. Recall that, for a cow chosen at random from the region, the probability that the test is positive is equal to 0.02. We denote by $X$ the random variable that associates to a sample of 100 cows from the region chosen at random the number of cows presenting a positive test in this sample. a. What is the probability distribution followed by the random variable $X$? Justify the answer and specify the parameters of this distribution. b. Calculate the probability that in a sample of 100 cows, there are exactly 3 cows presenting a positive test. Give a result to $10^{-3}$ near. c. Calculate the probability that in a sample of 100 cows, there are at most 3 cows presenting a positive test. Give a result to $10^{-3}$ near.
We now choose a sample of $n$ cows from this region, $n$ being a non-zero natural integer. We admit that this choice can be treated as a draw with replacement. Determine the minimum value of $n$ so that the probability that there is, in the sample, at least one cow tested positive, is greater than or equal to 0.99.
Paratuberculosis is an infectious digestive disease that affects cows. It is caused by the presence of a bacterium in the cow's intestine.
A study is conducted in a region where $0.4\%$ of the cow population is infected.
There is a test that reveals the immune response of an organism infected by the bacterium. The result of this test can be either ``positive'' or ``negative''.
A cow is chosen at random from the region. Given the characteristics of the test, we know that:
\begin{itemize}
\item If the cow is affected by the infection, the probability that its test is positive is 0.992;
\item If the cow is not affected by the infection, the probability that its test is negative is 0.984.
\end{itemize}
We denote by $I$ the event ``the cow is affected by the infection'' and $T$ the event ``the cow presents a positive test''. We denote by $\bar{I}$ the complementary event of $I$ and $\bar{T}$ the complementary event of $T$.
\textbf{Part A}
\begin{enumerate}
\item Reproduce and complete the weighted tree below modelling the situation.
\item
a. Calculate the probability that the cow is not affected by the infection and that its test is negative. Give the result to $10^{-3}$ near.\\
b. Show that the probability, to $10^{-3}$ near, that the cow presents a positive test is approximately equal to 0.020.\\
c. The ``positive predictive value of the test'' is the probability that the cow is affected by the infection given that its test is positive. Calculate the positive predictive value of this test. Give the result to $10^{-3}$ near.\\
d. The test gives incorrect information about the cow's state of health when the cow is not infected and presents a positive test result or when the cow is infected and presents a negative test result. Calculate the probability that this test gives incorrect information about the cow's state of health. Give a result to $10^{-3}$ near.
\end{enumerate}
\textbf{Part B}
\begin{enumerate}
\setcounter{enumi}{2}
\item When a sample of 100 cows is chosen at random from the region, this choice is treated as a draw with replacement. Recall that, for a cow chosen at random from the region, the probability that the test is positive is equal to 0.02. We denote by $X$ the random variable that associates to a sample of 100 cows from the region chosen at random the number of cows presenting a positive test in this sample.\\
a. What is the probability distribution followed by the random variable $X$? Justify the answer and specify the parameters of this distribution.\\
b. Calculate the probability that in a sample of 100 cows, there are exactly 3 cows presenting a positive test. Give a result to $10^{-3}$ near.\\
c. Calculate the probability that in a sample of 100 cows, there are at most 3 cows presenting a positive test. Give a result to $10^{-3}$ near.
\item We now choose a sample of $n$ cows from this region, $n$ being a non-zero natural integer. We admit that this choice can be treated as a draw with replacement. Determine the minimum value of $n$ so that the probability that there is, in the sample, at least one cow tested positive, is greater than or equal to 0.99.
\end{enumerate}