Multiple choice

A die is thrown $2n + 1$ times. The probability of getting $1$ or $3$ or $4$ atmost $n$ times is

  1. $\dfrac12$
  2. $\dfrac1n$
  3. $\dfrac{n}{(2n +1)}$
  4. $\dfrac1{(2n + 1)}$
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A Correct answer
Explanation

This is a binomial distribution problem with p = 3/6 = 1/2. For a symmetric distribution with 2n+1 trials, the probability of success in at most n trials is exactly 1/2 because the probability of success in at most n trials equals the probability of success in at least n+1 trials.

AI explanation

In 2n plus 1 throws of a die, the probability of getting 1, 3 or 4 at most n times equals the probability of getting 5, 2 or 6 at least n plus 1 times, making the two events symmetric and mutually exclusive. This symmetry arises because the probability of rolling 1, 3 or 4 is 1/2, exactly matching the probability of rolling 2, 5 or 6. Therefore, the required probability is exactly 1/2.

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