Answer:
Explanation:
Hello!
Any medical test used to detect certain sicknesses have several probabilities associated with their results.
Positive (test is +) ⇒ P(+)
True positive (test is + and the patient is sick) ⇒ P(+ ∩ S)
False-positive (test is + but the patient is healthy) ⇒P(+ ∩ H)
Negative (test is -) ⇒ P(-)
True negative (test is - and the patient is healthy) ⇒ P(- ∩ H)
False-negative (test is - but the patient is sick) ⇒ P(- ∩ S)
The sensibility of the test is defined as the capacity of the test to detect the sickness in sick patients (true positive rate).
⇒ P(+/S) = P(+ ∩ S)
P(S)
The specificity of the test is the capacity of the test to have a negative result when the patients are truly healthy (true negative rate)
⇒ P(-/H) = P(- ∩ H)
P(H)
For this particular blood disease the following probabilities are known:
1% of the population has the disease: P(S)= 0.01
95% of those who are sick, test positive for it: P(+/S)= 0.95 (sensibility of the test)
2% of those who don't have the disease, test positive for it: P(+/H)= 0.02
The probability of a person having the blood sickness given that the test was positive is:
P(S/+)= P(+ ∩ S)
P(+)
The first step you need to calculate the intersection between both events + and S, for that you will use the information about the sickness prevalence in the population and the sensibility of the test:
P(+/S) = P(+ ∩ S)
P(S)
P(+/S)* P(S) = P(+ ∩ S)
P(+ ∩ S) = 0.95*0.01= 0.0095
The second step is to calculate the probability of the test being positive:
P(+)= P(+ ∩ S) + P(+ ∩ H)
Now we know that 1% of the population has the blood sickness, wich means that 99% of the population doesn't have it, symbolically: P(H)= 0.99
Then you can clear the value of P(+ ∩ H):
P(+/H) = P(+ ∩ H)
P(H)
P(+/H)*P(H) = P(+ ∩ H)
P(+ ∩ H) = 0.02*0.99= 0.0198
Next you can calculate P(+):
P(+)= P(+ ∩ S) + P(+ ∩ H)= 0.0095 + 0.0198= 0.0293
Now you can calculate the asked probability:
P(S/+)= P(+ ∩ S) = 0.0095 = 0.32
P(+) 0.0293
I hope it helps!