Multiple choice

$3$ players A,B & C toss a coin cyclically in the order (that is A, B, C, A, B, C, A, B.....) till a head shows. Let $p$ be the probability that the coin shows a head. Let $\alpha ,\beta\ &\ \gamma $ be respectively the probabilities that A, B and C gets the first head. Determine $\alpha $ (in terms of p)

  1. $\displaystyle \frac{(1-p)p}{1-(1-p)^3}$
  2. $\displaystyle \frac{p}{1-(1-p)^3}$
  3. $\displaystyle \frac{(1-p)^2p}{1-(1-p)^3}$

  4. none
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B Correct answer
Explanation

Let p be the probability of head. A wins if head occurs on 1st, 4th, 7th... toss. Probability = p + (1-p)^3 * p + (1-p)^6 * p + ... This is a geometric series with first term p and common ratio (1-p)^3. Sum = p / (1 - (1-p)^3).

AI explanation

Player A can get the first head on the first toss, with probability p, or on the fourth toss after all three players get tails, with probability (1-p)^3 * p, and so on. This forms an infinite geometric series: p + (1-p)^3 * p + (1-p)^6 * p. Using the sum of an infinite geometric progression formula, the first term is p and the common ratio is (1-p)^3. The sum is p divided by (1 - (1-p)^3), which is the probability alpha.

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