Multiple choice

A coin is tossed $5$ times. What is the probability that tail appears an odd number of times?

  1. $\cfrac{3}{5}$
  2. $\cfrac{2}{15}$
  3. $\cfrac{1}{2}$
  4. $\cfrac{1}{3}$
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C Correct answer
Explanation

For any number of tosses n, the probability of getting an odd number of tails is 1/2, because the binomial expansion of (p+q)^n and (p-q)^n shows that the sum of odd-indexed terms equals the sum of even-indexed terms when p=q=1/2.

AI explanation

Using the binomial distribution formula, the probability of getting an odd number of tails is the sum of the probabilities of getting exactly 1, 3, or 5 tails. For a fair coin tossed 5 times, this sum is 5C1/32 + 5C3/32 + 5C5/32, which equals (5 + 10 + 1)/32. Since (1/2)^5 is 1/32, the total probability is 16/32, which equals 1/2.

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