Fast Baseball Pitcher
A baseball team hires a pitcher who throws so fast that no one could ever hit his pitches. Instead, he either strikes out or walks every batter. He strikes out any given batter with probability \(p.\) To have an earned run average less than \(1.00,\) what is the minimum possible value of \(p?\)
Details: In baseball, there are 9 innings. An earned run average is the number of runs given up per 9 innings. After 3 people are walked, each additional walk scores a run. After 3 strike outs, the inning ends.
Note: While the majority of this problem can be solved with combinatorics, you may need a computing device for the last step.