Problems from 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?

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Development:

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, $$5,8\%$$.

Solution:

The probability is $$0,058$$.

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