# 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 posi - Mathematics and Statistics

Sum

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 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 tested positive. What is the probability that the subject has serious form of the disease?

#### Solution

Let event A1: Disease in serious form,

event A2: Disease in mild form

event A3: Subject does not have disease,

event B: Subject tests positive.

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

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.

∴ "P"("B"/"A"_1) = 0.9, "P"("B"/"A"_2) = 0.6, "p"("B"/"A"_3) = 0.1

P(B) = "P"("A"_1) "P"("B"/"A"_1) + "P"("A"_2) "P"("B"/"A"_2) + "P"("A"_3) "P"("B"/"A"_3)

= 0.02 × 0.9 + 0.1 × 0.6 + 0.88 × 0.1

= 0.166

Required probability = "P"("A"_1/"B")

By Baye’s theorem

"P"("A"_1/"B") = ("P"("A"_1) "P"("B"/"A"_1))/("P"("B"))

= (0.9 xx 0.02)/0.166

= 0.108

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