Question

Doctors have devised a test for leptospirosis that has the following property: For any person suffering from lepto, there is a 90% chance of the test returning positive. For a person not suffering from lepto, there is an 80% chance of the test returning negative. It is known that 10% of people who go for testing have lepto. If a person who gets tested gets a +ve result for lepto (as in, the test result says they have got lepto), what is the probability that they actually have lepto?

• A. 1/2
• B. 1/3
• C. 1/4
• D. 1/7

Answer 1

The correct option is B 1/3

Let us draw the possibilities in this scenario.
Prob (patient having lepto) = 0.9
Prob (patient not having lepto) = 0.1
Given that patient has lepto, Prob (test being positive) = 0.9
Given that patient has lepto, (Prob test being negative) = 0.1
Given that patient does not have lepto, Prob (test being negative) = 0.8
Given that patient does not have lepto, Prob (test being positive) = 0.2
Now, we are told that the test turns positive. This could happen under two scenarios - the patient has lepto and the test turns positive and patient does not have lepto and the test turns positive.
Probability of test turning positive = 0.9 $\times$ 0.1 + 0.9 $\times$ 0.2 = 0.27.
Now, we have not been asked for the probability of test turning positive. We are asked for the probability of a patient having lepto given that he/she tests positive. So, the patient has already tested positive. So, this 0.27 includes the set of universal outcomes. Or, this 0.27 sits in the denominator.
Within this 0.27, which subset was the scenario that the patient does indeed have lepto?
This is the key question. This probability is 0.1 $\times$ 0.9 = 0.09. So, the required probability = 0.09/0.27 = 1/3
So, if a patient tests positive, there is a 1 in 3 chance of him/her having lepto. This is the key reason that we need to be careful with medical test results.