Activity › Discussion › Math › Probability › Reply To: Probability

::
To find the probability that a person actually has the disease given that they tested positive, we can use Bayes’ theorem. Let’s define the following probabilities:
P(D) = Probability of having the disease = 0.01 (1% of individuals)
P(PosD) = Probability of testing positive given that a person has the disease = 0.95 (95% accuracy)
P(Pos¬D) = Probability of testing positive given that a person does not have the disease = 0.02 (2% false positive rate)
We want to calculate P(DPos), which is the probability of having the disease given that the person tested positive. According to Bayes’ theorem:
P(DPos) = (P(PosD) * P(D)) / P(Pos)
To calculate P(Pos), the probability of testing positive, we need to consider both scenarios: the person having the disease and testing positive, and the person not having the disease but testing positive.
P(Pos) = P(PosD) * P(D) + P(Pos¬D) * P(¬D)
P(¬D) = Probability of not having the disease = 1 – P(D) = 0.99 (99% of individuals)
Now we can substitute these values into the equation:
P(DPos) = (P(PosD) * P(D)) / (P(PosD) * P(D) + P(Pos¬D) * P(¬D))
P(DPos) = (0.95 * 0.01) / (0.95 * 0.01 + 0.02 * 0.99)
Simplifying the equation:
P(DPos) = 0.0095 / (0.0095 + 0.0198)
P(DPos) ≈ 0.324
Therefore, the probability that a person actually has the disease given that they tested positive is approximately 0.324 or 32.4%.