Final answer:
The probability that a person actually has the disease given a positive test result can be calculated using Bayes' Theorem, which takes into account the test's sensitivity and specificity as well as the prevalence of the disease.
Step-by-step explanation:
To answer the question regarding the probability that a person has the disease given a positive blood test result, we can use Bayes' Theorem. This theorem combines the individual probabilities of testing positive or negative with the overall prevalence of the disease to arrive at the probability we are seeking.
Let's define the following events:
- D = person has the disease
- ~D = person does not have the disease
- + = test outcome is positive
- - = test outcome is negative
From the question we know:
- P(D) = 0.01 (prevalence of the disease)
- P(+|D) = 0.95 (sensitivity of the test)
- P(-|~D) = 0.98 (specificity of the test)
We need to find P(D|+), the probability that the person has the disease given a positive test result. Bayes' Theorem formula in this context is:
P(D|+) = \(\fracD) \cdot P(D)P(+\)
Where P(+|~D) represents the probability of a false positive, which is 1 - P(-|~D) = 0.02, and P(~D) is the probability of not having the disease, which is 1 - P(D) = 0.99.
Substituting the known values:
P(D|+) = \(\frac{0.95 \cdot 0.01}{0.95 \cdot 0.01 + 0.02 \cdot 0.99}\)
Calculating this gives the probability that a person actually has the disease given a positive test result.