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Q.
An unbiased die is rolled until a number greater than $4$ appears. The probability that an even number of trials are needed, is
NTA AbhyasNTA Abhyas 2022
Solution:
Probability of success $P\left(s\right)=p=\frac{2}{6}=\frac{1}{3}$
Probability of failure $P\left(f\right)=q=1-\frac{1}{3}=\frac{2}{3}$
The probability that success occurs in even number of trials
$=P\left(f s\right)+P\left(f f f s\right)+P\left(f f f f f s\right)+.....$
$=qp+q^{3}p+q^{5}p+\ldots =\frac{q p}{1 - q^{2}}$
$=\frac{\frac{2}{3} \times \frac{1}{3}}{1 - \left(\frac{2}{3}\right)^{2}}=\frac{\frac{2}{9}}{1 - \frac{4}{9}}=\frac{2}{9}\times \frac{9}{5}=\frac{2}{5}$