1. Problem 9.5 A stick is broken at random into two pieces. You bet on the ratio of the length of the longer piece to the length of the smaller piece. You receive $k if the ratio is between k and k + 1 for some k with 1 ≤ k ≤ m −1, while you receive $m if the ratio is larger than m. Here m is a given positive integer. What should be your stake to makethisafair bet? Verify that your stake should be $2[1 + 1 2

+···+ 1 m+1

−1] (this amount is approximately equal to

$2[ln(m + 1) +γ −1+ 1 2(m+1)

] for m large, where γ = 0.57722...is Euler’s constant).