A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is [JEE Main 2024, …

Probability of getting \(2\) = \(\frac{1}{6}\)

Probability of getting other than \(2\) = \(\frac{5}{6}\)

Required Probability \(=\frac{5}{6}\times \frac{1}{6}+\frac{5}{6}\times \frac{5}{6}\times \frac{5}{6}\times \frac{1}{6}+\ldots \ldots \ldots\)

\(=\frac{5}{36}\left[1+{\left(\frac{5}{6}\right)}^{2}+{\left(\frac{5}{6}\right)}^{4}+\ldots \ldots \ldots \right]=\frac{5}{36}\times \frac{1}{1−\frac{25}{36}}=\frac{5}{11}\)