Multiple choice

A dice is thrown $2n + 1$ times, $\displaystyle n : \epsilon : N$. The probability that faces with even numbers show odd number of times is

  1. $\displaystyle \frac {2n + 1}{4n + 3}$
  2. less than $\displaystyle \frac {1}{2}$
  3. greater than $\displaystyle \frac {1}{2}$

  4. none of these
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D Correct answer
AI explanation

Using the binomial distribution, the probability of an even number showing an odd number of times is the sum from k=1 to 2n+1 of the combination of 2n+1 choose k times (1/2)^k times (1/2)^(2n+1-k). This simplifies to (1/2)^(2n+1) times the sum of odd-indexed binomial coefficients for 2n+1. The sum of the odd binomial coefficients equals the sum of the even binomial coefficients, which is exactly 2^(2n). Therefore, the probability is 2^(2n) divided by 2^(2n+1), which equals exactly 1/2. Because the probability is exactly 1/2, it is neither less than nor greater than 1/2.

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