First slide

Theorems of probability

Question

$A biased coin with probability P, 0 < P < 1 of head, is tossed until a head appears for the$ $first time. If the probability that the number of tosses required is even is \frac{2}{5}, then the value of P is$

Moderate

Solution

$\begin{matrix} Given probability of head = P \\ G i v e n \frac{2}{5} = (1 - P) P + (1 - P) (1 - P) (1 - P) P + \dots . \\ = (1 - P) P + [1 + (1 - P)^{2} + (1 - P)^{4} + \dots . .] \\ \Rightarrow \frac{2}{5} = (1 - P) P [\frac{1}{1 - (1 - P)^{2}}] (∵ In G.Pseries, S_{\infty} = \frac{a}{1 - r}) \\ \Rightarrow \frac{2}{5} = \frac{P - P^{2}}{2 P - P^{2}} \\ \Rightarrow 5 P - 5 P^{2} = 4 P - 2 P^{2} \\ \Rightarrow P = 0 (or) P = \frac{1}{3} \\ ∴ P = \frac{1}{3} = 0.33 \end{matrix}$

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