Question

A pitcher throws a strike with probability 75% and a ball with probability 25%. Although reality is probably more complicated, we'll assume that the result of each pitch is independent from another. In other words, it doesn't matter what the past history of strikes/balls has been, the probability the next pitch is a strike is 75%.

Let's imagine the pitcher is currently practicing on an empty plate (no batter).

a. What is the probability that the first two pitches are both strikes?

b. When the previous two pitches have both been strikes, what is the probability the next pitch will be a strike?

c. What is the probability of getting a strike on the first pitch or second pitch? Remember that "or" in probability really means "at least one of", so this is asking "what is the probability that at least one of the first two pitches are strikes".

d. What is the probability that the pitcher will be able to throw 9 strikes in a row (enough to end an inning in the fastest way possible).

e. Using the complement rule, find the probability that the pitcher will have at least one strike in his first 6 pitches (note: the event "at least one" is the complement of "zero").

f. Find the probability of the following sequence of pitches (S = strike, B = ball): SBBSBS

g. (Bonus) Find the probability of a "strike out", i.e., that 3 strikes occur before 4 balls. Hint: write out all possible sequences of strikes/balls that result in a "strike out", find the probability of each sequence, then appropriately combine the results. Sanity check: I found the result close to 96%.

(If you could help with coding to figure out the answers to these questions as well)

Answer 1