Multiple choice

A coin is tossed $n$ times, the probability that head will turn up an even number of times is

  1. $ \displaystyle \frac{n+1}{2n} $
  2. $ \displaystyle \frac{n}{n+1} $
  3. $ \displaystyle \frac{1}{2} $
  4. $\displaystyle 2^{n-1}$
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C Correct answer
Explanation

The probability of getting an even number of heads in n tosses of a fair coin is given by the sum of binomial probabilities for even k, which is always 1/2 for any n >= 1.

AI explanation

Using the binomial distribution, the probability of getting an even number of heads is the sum for all even integers k from 0 to n of nCk multiplied by (1/2) raised to the power of n. This sum equals exactly half of the total sum of the binomial expansion for (1/2 + 1/2) raised to the power of n. Therefore, the probability is 1/2.

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