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 chance of making the shot. If denotes the expected number of points Bob will make and the number of points Bob is most likely to make, then for relatively prime positive integers and Find

Solution

Firstly, the expected number of shots Bob makes is Now, the number of shots Bob is most likely to make is the largest integer such that \begin{align*} \left(\frac{35}{100}\right)^a\left(\frac{65}{100}\right)^{97-a}\binom{97}{a}&\geq\left(\frac{35}{100}\right)^{a-1}\left(\frac{65}{100}\right)^{98-a}\binom{97}{a-1}\implies\\ 35(98-a)&\geq65(a)\implies\\ 3430&\geq100a\implies\\ 34&\geq a. \end{align*} So, Bob is most likely to make shots. Therefore, the answer is

~SMO_Team