Multiple choice

A coin is tossed $\left (2n + 1 \right )$ times, the probability that head appear odd number of times is

  1. $\displaystyle \frac{n}{2n+1}$
  2. $\displaystyle \frac{n+1}{2n+1}$
  3. $\displaystyle \frac{1}{2}$

  4. None of these
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C Correct answer
Explanation

For a fair coin tossed an odd number of times, the probability of getting an odd number of heads is exactly 1/2 due to symmetry.

AI explanation

By the properties of the binomial distribution for independent Bernoulli trials with a fair coin, the sum of the odd-indexed binomial coefficients is equal to the sum of the even-indexed binomial coefficients. The total probability of all possible outcomes is 1, so the probability of getting an odd number of heads is exactly half of the total probability. This means the probability is 1/2.

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