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GATE 2025 DA – Question 45

Probability and Statistics · Counting, probability axioms, conditional probability and Bayes theorem · 2 marks · Multiple choice

A random experiment consists of throwing 100 fair dice, each die having six faces numbered 1 to 6. An event $A$ represents the set of all outcomes where at least one of the dice shows a 1. Then, $P(A) =$

  1. 0
  2. 1
  3. $1 - \left(\frac{5}{6}\right)^{100}$
  4. $\left(\frac{5}{6}\right)^{100}$

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Correct answer: (C) $1 - \left(\frac{5}{6}\right)^{100}$

Explanation

The complement of $A$ is that no die shows a 1. Each die avoids a 1 with probability $\frac{5}{6}$ and the dice are independent, so the complement has probability $\left(\frac{5}{6}\right)^{100}$. So $P(A) = 1 - \left(\frac{5}{6}\right)^{100}$.