GATE CS 2017 Set 2 Q.26 (CS Q.26): Probability and Statistics

P and Q are considering to apply for a job. The probability that P applies for the job is $\frac{1}{4}$ , the probability that P applies for the job given that Q applies for the job is $\frac{1}{2}$ , and the probability that Q applies for the job given that P applies for the job is $\frac{1}{3}$ . Then the probability that P does not apply for the job given that Q does not apply for the job is

$\frac{4}{5}$

$\frac{5}{6}$

$\frac{7}{8}$

$\frac{11}{12}$

Official answer and solution

Official answer: A
From: Official GATE 2017 answer key

Solution

Use the definition of conditional probability to compute the joint probability $P(P \cap Q)$ and marginal probability $P(Q)$ , then evaluate $P(\neg P \mid \neg Q)$ .

Let $A$ denote the event that P applies for the job, and $B$ denote the event that Q applies for the job.
Given:

$P(A) = \frac{1}{4}$
$P(A \mid B) = \frac{1}{2}$
$P(B \mid A) = \frac{1}{3}$

Find $P(A \cap B)$ and $P(B)$ :

P(A \cap B) = P(B \mid A) \cdot P(A) = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12}

Using $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ :

P(B) = \frac{P(A \cap B)}{P(A \mid B)} = \frac{1/12}{1/2} = \frac{2}{12} = \frac{1}{6}

Now find $P(A \cup B)$ :

P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{1}{4} + \frac{1}{6} - \frac{1}{12} = \frac{3 + 2 - 1}{12} = \frac{4}{12} = \frac{1}{3}

Now compute the requested conditional probability $P(\neg A \mid \neg B)$ :

P(\neg A \mid \neg B) = \frac{P(\neg A \cap \neg B)}{P(\neg B)} = \frac{1 - P(A \cup B)}{1 - P(B)}

= \frac{1 - \frac{1}{3}}{1 - \frac{1}{6}} = \frac{\frac{2}{3}}{\frac{5}{6}} = \frac{2}{3} \times \frac{6}{5} = \frac{4}{5}

This corresponds to option A.

Final answer

Common trap

Assuming $P(\neg A \mid \neg B) = 1 - P(A \mid B)$ , which is a common fallacy in conditional probability.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2017 Computer Science and Information Technology (CS) Session 2
Question: 26 of 65, printed as CS Q.26
Section: Computer Science and Information Technology

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