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- Probability
- Bayes's theorem
- Ejercicios
Bayes's theorem
In a country there is a certain disease that affects one out of $145$ people. We have a test to detect that disease, but it is not completely reliable: if the individual has the disease, the test gives positive $96\%$ of the times, while if it does not have it, the test gives positive $6\%$ of the times. If a person takes the test and the result is positive: what is the probability that the test is mistaken?
Let's consider the following events: $E =$ "sick", and $S=\overline{E}=$ "healthy". On the other hand, $P =$ "positive result in the test", and $N=\overline{P}=$ "negative result".
We can make a tree, using all the probabilities that we have together with the ones that we can deduce: for example, if one out of the $145$ individuals who have the desease, then it means that $144$ out of $145$ are healthy. Remember that the probabilities of each of the possible events has to add up to $1$.
We are looking for $P(S/P)$. From Bayes' theorem, $$ P(S/P)=\dfrac{P(S)\cdot P(P/S)}{P(S)\cdot P(P/S)+ P(E)\cdot P(P/E)}$$
In our case, $$P(S/P)=\dfrac{\dfrac{144}{145}\cdot\dfrac{6}{100}}{\dfrac{144}{145} \cdot\dfrac{6}{100}+\dfrac{1}{145} \cdot\dfrac{96}{100}}= 0,058$$
In other words, $5,8\%$.
The probability is $0,058$.