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 $S = \overset{―}{E} =$ "healthy". On the other hand, $P =$ "positive result in the test", and $N = \overset{―}{P} =$ "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, $P (S / P) = \frac{P (S) \cdot P (P / S)}{P (S) \cdot P (P / S) + P (E) \cdot P (P / E)}$

In our case, $P (S / P) = \frac{\frac{144}{145} \cdot \frac{6}{100}}{\frac{144}{145} \cdot \frac{6}{100} + \frac{1}{145} \cdot \frac{96}{100}} = 0, 058$

In other words, $5, 8 %$ .

Solution:

The probability is $0, 058$ .

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