From Lucas Jaeger comes a “flute-iful” challenge:
You’re a contestant on the hit new game show, “You Bet Your Fife.” On the show, a random real number (i.e., decimals are allowed) is chosen between 0 and 100. Your job is to guess a value that is less than this randomly chosen number. Your reward for winning is a novelty fife that is valued precisely at your guess. For example, if the number is 75 and you guess 5, you’d win a \$5 fife, but if you’d guessed 60, you’d win a \\$60 fife. Meanwhile, a guess of 80 would win you nothing.
What number should you guess to maximize the average value of your fifing winnings?
For the number $n$ you choose, the probability of it winning is $1 - \dfrac{n}{100}$.
The expected value is therefore $\dfrac{100n - n^2}{100}$, which has a maximum value at $n = 50$.
The maximum average value is thus \$25, achieved by choosing \$50, which wins half the time.