Blood tests are sometimes used to indicate if a person has a particular disease - Leaving Cert Mathematics - Question 8 - 2014

To find the overall probability that a person tests positive for the disease, we can use the law of total probability:

$P(Positive\,test) = P(Tests\,positive | Has\,the\,disease) \cdot P(Has\,the\,disease) + P(Tests\,positive | Does\,not\,have\,the\,disease) \cdot P(Does\,not\,have\,the\,disease)$

Substituting the values:

$P(Positive\,test) = (0.99 \cdot 0.003) + (0.04 \cdot 0.997)$ $= 0.00297 + 0.03988$ $= 0.04285$

Thus, the probability that a person selected at random from the population tests positive for the disease is $0.04285$.

To find the probability that a person has the disease given they tested positive, we can use Bayes' theorem:

$P(Has\,the\,disease | Positive\,test) = \frac{P(Positive\,test | Has\,the\,disease) \cdot P(Has\,the\,disease)}{P(Positive\,test)}$

Substituting in the known values:

$P(Has\,the\,disease | Positive\,test) = \frac{0.99 \cdot 0.003}{0.04285}$ $= \frac{0.00297}{0.04285} \approx 0.0693$

Therefore, the probability that the person actually has the disease, given that they tested positive, is approximately $0.0693$, or to three significant figures, $0.0693$.