Multiple choice

A biased coin with probability p, 0 < p < 1, of head is tossed until a head appears for the first time. If the probability that the number of tossess required is even is 2/5, then p = .......

  1. $\dfrac{1}{2}$
  2. $\dfrac{1}{4}$
  3. $\dfrac{1}{6}$
  4. $\dfrac{1}{3}$
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D Correct answer
Explanation

The probability of the first head occurring on an even toss is p(1 - p) + p(1 - p)^3 + ... = (1 - p)/(2 - p). Setting this equal to 2/5 gives p = 1/3.

AI explanation

Let q be the probability of tails, where q = 1 - p. The probability that the first head appears on an even toss is the infinite geometric series p + p^3 + p^5 + ... which equals p / (1 - p^2) = p / (q x (1 + p)). Using the given probability of 2/5 and substituting 1 + p = 2 - q, we get p / (2q - q^2) = 2/5. Solving this quadratic equation 4q^2 - 13q + 6 = 0 yields q = 2/3, meaning the probability of a head is p = 1/3.

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