GATE 2026 DA Q.57: solution and answer

GATE 2026 · DAQ.57 · 2 marks · NAT

A clinic specializes in testing for a disease D. The result of the test can be either positive or negative.

A study revealed that if a person suffers from the disease D, the test result in that clinic comes out positive 80% of the time, and negative 20% of the time. If a person is not suffering from the disease D, the test comes out positive 10% of the time and negative 90% of the time. It is also known that among the general population, the disease D occurs in 30% of the individuals.

If a person tests positive for D in that clinic, the probability that he/she actually suffers from the disease D is ________. (Rounded off to two decimal places)

See the question in the paper

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Answer and solution

ANSWER

The probability that a person who tests positive actually has the disease is $\frac{24}{31} \approx 0.77$ .

Official answer: 0.76 to 0.78Source: Official answer key, GATE 2026 DA (IIT Guwahati)

Let $D$ be the event that the person has the disease and $+$ the event that the test is positive. The population rate gives $P (D) = 0.30$ , so $P (\overset{ˉ}{D}) = 0.70$ .
The test results given are $P (+ ∣ D) = 0.80$ (and $P (- ∣ D) = 0.20$ ), and $P (+ ∣ \overset{ˉ}{D}) = 0.10$ (and $P (- ∣ \overset{ˉ}{D}) = 0.90$ ).
By the law of total probability, $P (+) = P (+ ∣ D) P (D) + P (+ ∣ \overset{ˉ}{D}) P (\overset{ˉ}{D}) = 0.8 \times 0.3 + 0.1 \times 0.7 = 0.24 + 0.07 = 0.31.$
By Bayes' theorem, $P (D ∣ +) = \frac{P ( + ∣ D ) P ( D )}{P ( + )} = \frac{0.24}{0.31} = \frac{24}{31} \approx 0.7742.$
Rounded to two decimal places, $P (D ∣ +) = 0.77$ .

KEY IDEA

The question asks for $P (D ∣ +)$ while the data give $P (+ ∣ D)$ , so reverse the conditioning with Bayes' theorem. Find the denominator $P (+)$ first, using total probability over $D$ and $\overset{ˉ}{D}$ .

WATCH OUT FOR

Reporting $P (+ ∣ D) = 0.80$ as the answer, which confuses the two conditional probabilities. Another slip is dividing $0.24$ by $0.8 + 0.1$ and ignoring the prior weights $0.3$ and $0.7$ in the denominator.

From the GATE 2026 DA paper. Every question of this paper · Download the full paper · All 3 Conditional & Bayes questions, with notes

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