The question is from Permutation and Combination. This is about probability of an event. An application based question based on bayes theorem. This section hosts a number of questions which are on par with CAT questions in difficulty on CAT Permutation and Combination, and CAT Probability.

Question 15: 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?

- \\frac{7}{10}\\)
- \\frac{8}{11}\\)
- \\frac{1}{3}\\)
- \\frac{1}{2}\\)

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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 * 0.1 + 0.9 * 0.2 = 0.27.

Now, we have not been asked for the probability of test turning positive. We are asked for the probability of 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 * 0.9 = 0.09.

So, the required probability = \\frac{0.09}{0.27}\\) = \\frac{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.

The question is **"what is the probability that they actually have lepto?"**

Choice C is the correct answer.

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