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GATE 2026 DA – Question 19

Probability and Statistics · Counting, probability axioms, conditional probability and Bayes theorem · 1 mark · Multiple choice

Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \ldots, 2026\}$.

Which of the following is the probability that the product of all the elements of $M$ is even?

  1. $\frac{2^{1013}(2^{1013} - 1)}{2^{2026}}$
  2. $\frac{2^{1013}}{2^{2026}}$
  3. $\frac{2^{1013}(2^{1013} - 1)}{2^{2026} - 1}$
  4. $\frac{1}{2^{2026} - 1}$

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Correct answer: (C) $\frac{2^{1013}(2^{1013} - 1)}{2^{2026} - 1}$

Explanation

There are $2^{2026} - 1$ non-empty subsets, all equally likely. The product is odd only if every element is odd, and there are 1013 odd numbers in $S$, so there are $2^{1013} - 1$ non-empty subsets of odd numbers. The number of subsets with an even product is $(2^{2026} - 1) - (2^{1013} - 1) = 2^{2026} - 2^{1013} = 2^{1013}(2^{1013} - 1)$. So the probability is $\frac{2^{1013}(2^{1013} - 1)}{2^{2026} - 1}$.