By examining the chest X ray, the probability that TB is detected when a person is actually suffering is 0.99. The probability of an healthy person diagnosed to have TB is 0.001. In a certain city, 1 in 1000 people suffers from TB. A person is selected at random and is diagnosed to have TB. What is the probability that he actually has TB?

Question

Philoid · Accepted Answer

Let events E₁, E₂, E₃ be the following:

E₁ be the event that person has TB and E₂ be the event that the person does not have TB

Total persons = 1000

So,

Now, Let E be the event that the person is diagnosed to have TB

P(E|E₁) is the probability that TB is detected when a person is actually suffering

P(E|E₂) the probability of an healthy person diagnosed to have TB

So, P(E|E₁) = 0.99 and P(E|E₂) = 0.001

Now, we have to find the probability that the person actually has TB

We use Bayes’ theorem to find the probability of occurrence of an event A when event B has already occurred.

∴

P(E₁|E) is the probability that person actually has TB

[Divide by 9 both numerator and denominator]