# P and Q are playing a game in which they toss a biased coin alternately.the probability of Heads is 0.2. The game continous till head comes and person getting head wins out - find the probability that P wins if Q starts. Option 1) $\frac{1}{3}$ Option 2) $\frac{2}{3}$ Option 3) $\frac{5}{9}$ Option 4) $\frac{4}{9}$

Use the concept

Probability of occurrence of an event -

Let S be the sample space then the probability of occurrence of an event E is denoted by P(E) and it is defined as

$P\left ( E \right )=\frac{n\left ( E \right )}{n\left ( S \right )}$

$P\left ( E \right )\leq 1$

$P(E)=\lim_{n\rightarrow\infty}\left(\frac{r}{n} \right )$

- wherein

Where n repeated experiment and E occurs r times.

$P(P)=0.2\rightarrow winning$

$P(\bar{P})=0.8\rightarrow Loss$

$\therefore for\: winning \: Q$

$=0.2+0.8\times 0.8\times 0.2+-----\infty$

$=>\frac{0.2}{1-(0.8)^{2}}=\frac{0.2}{1-0.64}=\frac{0.2}{0.36}$

$\frac{20}{36}=\frac{5}{9}$

$\therefore winning\: chance\: for\: P=1-\frac{5}{9}$

$=\frac{4}{9}$

Option 1)

$\frac{1}{3}$

Option is incorrect

Option 2)

$\frac{2}{3}$

Option is incorrect

Option 3)

$\frac{5}{9}$

Option is incorrect

Option 4)

$\frac{4}{9}$

Option is correct

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