To find the probability that a pitcher will throw exactly 3 pitches in an at-bat, we can follow these steps:
### Step 1: Identify the frequencies and the total number of at-bats
From the table provided, the frequencies of pitches thrown are:
- 12 at-bats with exactly 1 pitch
- 16 at-bats with exactly 2 pitches
- 32 at-bats with exactly 3 pitches
- 12 at-bats with exactly 4 pitches
- 8 at-bats with exactly 5 pitches
### Step 2: Calculate the total number of at-bats
To find the total number of at-bats, we sum up all the frequencies:
[tex]\[
12 + 16 + 32 + 12 + 8 = 80
\][/tex]
So, the total number of at-bats is 80.
### Step 3: Identify the frequency of at-bats with exactly 3 pitches
From the table, we see that there are 32 at-bats where exactly 3 pitches were thrown.
### Step 4: Calculate the probability
The probability [tex]\( P(3) \)[/tex] that exactly 3 pitches are thrown in an at-bat is given by the ratio of the frequency of at-bats with exactly 3 pitches to the total number of at-bats:
[tex]\[
P(3) = \frac{\text{Frequency of 3 pitches}}{\text{Total number of at-bats}} = \frac{32}{80}
\][/tex]
### Step 5: Simplify the fraction
[tex]\[
\frac{32}{80} = 0.4
\][/tex]
### Conclusion
The probability that a pitcher will throw exactly 3 pitches in an at-bat is:
[tex]\[
P(3) = 0.4
\][/tex]
