GATE CS 2015 Set 3 Q.47 (CS Q.37): Probability and Statistics

Suppose $X_i$ for $i = 1, 2, 3$ are independent and identically distributed random variables whose probability mass functions are $\Pr[X_i = 0] = \Pr[X_i = 1] = 1/2$ for $i = 1, 2, 3$ . Define another random variable $Y = X_1 X_2 \oplus X_3$ , where $\oplus$ denotes XOR. Then

\Pr[Y = 0 \mid X_3 = 0]

is _____.

Your answer

Official answer and solution

Official answer: 0.75
From: Official GATE 2015 answer key

Solution

Condition on X3 = 0 and compute the probability of the product of independent Bernoulli variables.

We are given $Y = X_1 X_2 \oplus X_3$ where $X_1, X_2, X_3$ are i.i.d. Bernoulli $(1/2)$ random variables.

Given $X_3 = 0$ :

Y = (X_1 X_2) \oplus 0 = X_1 X_2

Thus:

\Pr[Y = 0 \mid X_3 = 0] = \Pr[X_1 X_2 = 0]

Since $X_1$ and $X_2$ are independent binary random variables:

\Pr[X_1 X_2 = 1] = \Pr[X_1 = 1] \times \Pr[X_2 = 1] = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 0.25

Therefore:

\Pr[X_1 X_2 = 0] = 1 - \Pr[X_1 X_2 = 1] = 1 - 0.25 = 0.75

Final answer

0.75

Common trap

Confusing XOR with OR, or forgetting to complement Pr[X1*X2 = 1].

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

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 3
Question: 47 of 65, printed as CS Q.37
Section: Computer Science and Information Technology

Report an error in this question

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.

Question

Solution

Source