Question

8) 2% of the population have a certain blood
disease of a serious form: 10% have it in a
mild form; and 88% don't have it at all. A
new blood test is developed; the probability
of testing positive is jo if the subject has the
serious form, 1 if the subject has the mild
form, and to if the subject doesn't have the
disease. A subject is tested positive. What is
the probability that the subject has serious
form of the disease?​

Answer 1

The probability oh having a disease is 0.108.

The question is incomplete.

The correct Question is - 2% of the population have a certain blood disease in a serious form; 10%  have it in a mild form; and 88% don’t have it at all. A new blood test is developed;  the probability of testing positive is 9/10 if the subject has the serious form, 6/10 if the  subject has the mild form, and 1/10 if the subject doesn’t have the disease. A subject is just tested positive. What is the probability that I have the serious form of the disease?

Let the person that has disease in serious form = A1

Let the person that has disease in mild form = A2

Let the person that doesn’t have a disease = A3

Let the test positive = B

Thus,

P(A1) = 0.02, P(A2) = 0.1, P(A3) = 0.88

P(B | A1) = 0.9, P(B | A2) = 0.6, P(B | A3) = 0.1

Using , the theorem of total probability,

P(B) = 0.9×0.02+0.6×0.1+0.1×0.88

= 0.166,

Now, according to Bayes’ Theorem -

P(A1 | B) = P(B | A1)P(A1)  /P(B)

=  0.9×0.02 / 0.166

= 0.108

Thus, the probability oh having a disease is 0.108.