2024 SSMO Tiebreaker Round Problems/Problem 2

Problem

Bob is attempting to shoot a 3-point throw. Bob attempts the basket 97 times. Each time, Bob has a $35\%$ chance of making the shot. If $S_1$ denotes the expected number of points Bob will make and $S_2$ the number of points Bob is most likely to make, then $|S_1-S_2| = \frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution