Question

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 serious form, 6/10 if the subject has mild form and 1/10 if the subject doesn't have the disease. If a subject is tested positive what is the probability that he/she has serious form of the disease?

Answer 1

Answer:

18/1000

=0.018

thus there is probability of 0.018 that he or she has serious form of disease

Answer 2

Given :

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.

• probability of testing positive is

9/10 if the subject has serious form,

6/10 if the subject has mild form and

1/10 if the subject doesn't have the disease

To find:

The probability that he/she has serious form of the disease if a subject is tested positive

Solution:

Probability that a person to be tested positive =

percentage of people having in a serious form x probablity that the subject test positive if he has serious form +

percentage of people having in a mild form x probablity that the subject test positive if he has mild form  +

percentage of people having no disease x probablity that the subject test positive if he no disease.

• P( testing positive) = $\frac{2}{100}$x $\frac{9}{10}$ + $\frac{10}{100}$x$\frac{6}{10}$ + $\frac{88}{100}$x$\frac{1}{10}$ = (18 + 60 + 88 )/1000

• P(testing positive) = 166/1000

Now probability that a subject has serious form of disease, if tested positive is .

• P(serious/positive) = p(serious ∩ positive )/ p( positive )

• P ( serious / positive) = $\frac{2}{100}$x $\frac{9}{10}$ / $\frac{166}{1000}$ = 18/166 = 0.108