GATE 2024 DA Q.58: solution and answer

GATE 2024 · DAQ.58 · 2 marks · NAT

Consider two events $T$ and $S$ . Let $\overset{ˉ}{T}$ denote the complement of the event $T$ . The probability associated with different events are given as follows: $P (\overset{ˉ}{T}) = 0.6$ , $P (S ∣ T) = 0.3$ , $P (S ∣ \overset{ˉ}{T}) = 0.6$ . Then, $P (T ∣ S)$ is ______ (rounded off to two decimal places).

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Answer and solution

ANSWER

$P (T ∣ S) = 0.25$

Official answer: 0.25Source: Official answer key, GATE 2024 DA (IISc Bengaluru)

Complement rule: $P (T) = 1 - P (\overset{ˉ}{T}) = 1 - 0.6 = 0.4$ .
Total probability over the partition ${T, \overset{ˉ}{T}}$ : $P (S) = P (S ∣ T) P (T) + P (S ∣ \overset{ˉ}{T}) P (\overset{ˉ}{T})$ .
$P (S) = 0.3 \times 0.4 + 0.6 \times 0.6 = 0.12 + 0.36 = 0.48$ .
Bayes' theorem: $P (T ∣ S) = \frac{P ( S ∣ T ) P ( T )}{P ( S )} = \frac{0.12}{0.48}$ .
$P (T ∣ S) = \frac{0.12}{0.48} = 0.25$ , which is already exact to two decimal places.

KEY IDEA

The question gives $P (\overset{ˉ}{T})$ and the likelihoods $P (S ∣ T)$ , $P (S ∣ \overset{ˉ}{T})$ . Get the prior $P (T)$ from the complement rule, get $P (S)$ by total probability over $T$ and $\overset{ˉ}{T}$ , then reverse the conditioning with Bayes' theorem.

WATCH OUT FOR

Using $P (\overset{ˉ}{T}) = 0.6$ as if it were $P (T)$ . That gives $P (S) = 0.3 \times 0.6 + 0.6 \times 0.4 = 0.42$ and $P (T ∣ S) = 0.18/0.42 \approx 0.43$ . Another slip is dividing by $P (S ∣ T)$ or by $P (T)$ instead of $P (S)$ .

From the GATE 2024 DA paper. Every question of this paper · Download the full paper · All 3 Conditional & Bayes questions, with notes

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