# Math Help - probability problem

1. ## probability problem

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

(a) The batter walks on the fourth pitch.

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

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

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

(c) The batter walks

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

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

2. In order for the batter to be walked on the sixth pitch it must be true that among the first five pitches there must be two strikes and three balls while the sixth pitch must be a ball. Here is an example $\underbrace {BSBSB}_5\boxed{B}$.
But the first five places can be rearranged in ${5 \choose 3}$ ways.
Thus the probability is ${5 \choose 3} \left( {\frac{1}{2}} \right)^6$.

3. why is it not $\frac{5!}{3!}$? Doesnt order matter?

4. also for (c), isnt the probability 1/2 from the law of large numbers?

5. Originally Posted by shilz222
why is it not $\frac{5!}{3!}$? Doesnt order matter?
How many ways are there to arrrange the string "SSBBB"?
It is not $\frac {5!} {3!}$.

For (c): How many ways are there to arrrange the string "SSBBBB"?
Each of those strings has a probiblity $\left( {\frac{1}{2}} \right)^6$

6. $\binom{5}{3}$ and $\binom{6}{4}$ respectively.

7. and for part (d) the batter scores implies that he walks (assuming he does not hit the ball) right? so how would we do this?

8. Originally Posted by shilz222
and for part (d) the batter scores implies that he walks (assuming he does not hit the ball) right? so how would we do this?
Sorry, but I know so little about sports is general and baseball in particular, that I cannot answer your question.

9. yeah what that does mean exactly?