From Brad Fortner comes a puzzle for the cleverest among us:
You are very clever when it comes to solving Riddler Express puzzles. You are so clever, in fact, that you are in the top 10 percent of solvers in Riddler Nation (which, as you know, has a very large population). You don’t know where in the top 10 percent you are — in fact, you realize that you are equally likely to be anywhere in the topmost decile. Also, no two people in Riddler Nation are equally clever.
One Friday morning, you walk into a room with nine members randomly selected from Riddler Nation. What is the probability that you are the cleverest solver in the room?
Each of the nine members randomly has a 10% chance of being in the top 10 percent of solvers, and of that a 50% chance of being more clever than you.
Since each member has a 5% chance of being more clever than you, the pobability that you are the cleverest solver is:
$$\left(1 - 0.05\right) ^ 9 \approx 63.02 \%$$