\[ p(k) = \begin{cases} \frac{1}{5^k} & k = 1, 2, 3, \dots \\ 0 & \text{otherwise} \end{cases} \]
The probability that you have a rotten day is 10%. If your days are independent, then the number of rotten days you have in a week, call it \(R\), it binomially distributed with \(n = 7\) and \(p = 0.1\). Find \(P(R \geq 1)\).
A fair coin is tossed forever until both heads and tails are seen. What is the expected number of flips?
Try to evaluate this sum:
\[\sum_{k = 0}^\infty \frac{k^2}{k!}\]