Suppose that there are $\displaystyle 2 $ teams: $\displaystyle A $ and $\displaystyle B $. Team $\displaystyle A $'s pitcher throws a strike $\displaystyle 50 \% $ of the time and a ball $\displaystyle 50 \% $ of the time (successive pitches are independent from one another), and the pitcher never hits the batter. Knowing this, Team $\displaystyle B $'s manager instructs the first batter not to swing at anything. Calculate the following probabilities:

(a) The batter walks on the fourth pitch.

So $\displaystyle P \{\text{batter walks on fourth pitch} \} = \left(\frac{1}{2} \right)^4 = \frac{1}{16} $

(b) The batter walks on the sixth pitch (so two of the first five must be strikes).

So $\displaystyle P \{\text{batter walks on sixth pitch} \} = \frac{5 \cdot 4}{2^{5}} = \frac{20}{32} = \frac{5}{8} $

(c) The batter walks

So this is the same as $\displaystyle P \{\text{pitcher throws four balls in a row} \} = \frac{1}{16} $

(d) The first batter scores while no one is out. Not sure how to do this one.