Question

The probability that a person with certain symptoms has hepatitis is 0.70.7. The blood test used to confirm this diagnosis gives positive results for 9292​% of people with the disease and 55​% of those without the disease. What is the probability that an individual who has the symptoms and who reacts positively to the test actually has​ hepatitis?

Answer 1

Answer:

Therefore the probability that an individual who has the symptoms and reacts positively to test actually positively is 0.98.

Step-by-step explanation:

Bayes' Theorem:

$P(A|B)=\frac{P(B|A).P(A)}{P(B)}$

A,B = events

P(A|B)= probability of A given B is true

P(A),P(B)= the independent probabilities of A and B.

In this case we use Bayes' Theorem  (spacial case)

$P(F|E)=\frac{P(F).P(E|F)}{P(F).P(E|F)+P(F').P(E|F')}$

F= a person with symptoms has hepatitis

F'=  a person with symptoms does have not hepatitis

E= Blood test positive.

P(F)= 0.7

P(F') = 1- 0.7 = 0.3

P(E|F)= 92% = 0.92

P(E|F')= 5%=0.05

$P(F|E)= \frac{0.7 \times 0.92}{(0.7 \times 0.92)+(0.3 \times 0.05)}$

            ≈0.98

Therefore the probability that an individual who has the symptoms and reacts positively to test actually positively is 0.98.