Multiple choice

A symmetric die is thrown $(2n+1)$ times. The probability of getting a prime score on the upturned face at most $n$ times is

  1. $\displaystyle\frac{1}{2}$
  2. $\displaystyle\frac{1}{3}$
  3. $\displaystyle\frac{1}{4}$
  4. $\displaystyle\frac{2}{3}$
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A Correct answer
Explanation

Prime scores on a die are 2, 3, 5 (3 outcomes). Probability of prime = 3/6 = 1/2. In (2n+1) trials, the probability of getting at most n successes in a symmetric binomial distribution is 1/2 due to symmetry.

AI explanation

When a symmetric die is thrown an odd number of times, specifically 2n+1 times, the binomial distribution of getting a prime score versus a non-prime score is symmetric. Because a standard six-sided die has 3 prime faces (2, 3, 5) and 3 non-prime faces (1, 4, 6), the probability of getting a prime score on any throw is 1/2. Due to this symmetry, the probability of getting at most n prime scores exactly equals the probability of getting more than n prime scores. Therefore, the probability of getting a prime score at most n times is exactly 1/2.

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