Puzzles

The Puzzle Inspired by true events Alice is assigned to be the 56th passenger to board a full plane with 60 seats. However, a panic causes all the passengers–including Alice–to arrange themselves radomly in line to board. As Alice was originally 56th, she decides that she would be happy as long as passengers with the assigned spots 57, 58, 59, and 60 are not in front of her. What is the probability that Alice will be happy?...

I was recently doing a probability puzzle that I can’t quite remember the context of, but I came across the answer that the probability would be $$\mathbb{P}(X) = n p^n \; \quad \forall \: n\in\mathbb{N}, p \in [0,1].$$ But this is obviously wrong! Plug in $p=.9, n=2$, and you get that $\mathbb{P}(X) = 1.62$. Thaat’s not how probability works! However, for $p=0.5$, $\mathbb{P}(X)$ will remain $\leq 1$ for all $n \in \mathbb{N}$....