JEE Advance - Mathematics (1998 - No. 8)
Three players, $$A,B$$ and $$C,$$ toss a coin cyclically in that 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 $$\alpha ,\,\,\,\beta $$ and $$\gamma $$ be, respectively, the probabilities that $$A, B$$ and $$C$$ gets the first head. Prove that $$\beta = \left( {1 - p} \right)\alpha $$ Determine $$\alpha ,\beta $$ and $$\gamma $$ (in terms of $$p$$).
$$\alpha = \frac{p}{1 - (1-p)^3}, \beta = \frac{(1-p)p}{1 - (1-p)^3}, \gamma = \frac{p(1-p)^2}{1 - (1-p)^3}$$
$$\alpha = \frac{1}{3}, \beta = \frac{1}{3}, \gamma = \frac{1}{3}$$
$$\alpha = p, \beta = (1-p)p, \gamma = (1-p)^2 p$$
$$\alpha = \frac{p}{1 - (1-p)}, \beta = \frac{(1-p)p}{1 - (1-p)}, \gamma = \frac{p(1-p)^2}{1 - (1-p)}$$
$$\alpha = \frac{1}{1 - (1-p)^3}, \beta = \frac{(1-p)}{1 - (1-p)^3}, \gamma = \frac{(1-p)^2}{1 - (1-p)^3}$$
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