You are the manager for the New York Frets, a baseball team that has woefully underperformed this season. In an effort to right the ship, you are tinkering with the batting order.
Eight of your nine batters are “pure contact” hitters. One-third of the time, each of them gets a single, advancing any runners already on base by exactly one base. (Your team is very slow on the base paths. That means no one is fast enough to score from first or second base on a single—only from third). The other two-thirds of the time, they record an out, and no runners advance to the next base.
Your ninth batter is the slugger. One-tenth of the time, he hits a home run. But the remaining nine-tenths of the time, he strikes out.
Your goal is to score as many runs as possible, on average, in the first inning. Where in your lineup (first, second, third, etc.) should you place your home run slugger?
I ran code that, for each of the nine unique lineups, simulated cycling through the lineup, the hitters' probabilities, scoring runs, and accumulating 3 outs before terminating.
For each slugger position, I averaged the total number of home runs over 100,000,000 trials.
The slugger should be the fourth batter.