Certainly! Let's solve this problem step by step.
### Step 1: Identify the Given Data
We are given the frequency of pitches thrown in all the at-bats during a baseball game:
- 1 pitch: 12 times
- 2 pitches: 16 times
- 3 pitches: 32 times
- 4 pitches: 12 times
- 5 pitches: 8 times
### Step 2: Calculate the Total Number of At-Bats
To find the total number of at-bats, we need to sum up the frequencies of each type of pitch count:
[tex]\[
12 + 16 + 32 + 12 + 8
\][/tex]
### Step 3: Perform the Summation
[tex]\[
12 + 16 + 32 + 12 + 8 = 80
\][/tex]
So, the total number of at-bats is [tex]\( 80 \)[/tex].
### Step 4: Identify the Frequency of the Event of Interest
We are interested in finding the probability that exactly 3 pitches are thrown in an at-bat. The frequency for exactly 3 pitches is:
[tex]\[
32
\][/tex]
### Step 5: Calculate the Probability
The probability of an event is given by the ratio of the frequency of that event to the total frequency of all possible events. In this case, the probability that a pitcher will throw exactly 3 pitches in an at-bat is:
[tex]\[
P(3) = \frac{\text{Frequency of 3 pitches}}{\text{Total number of at-bats}} = \frac{32}{80}
\][/tex]
### Step 6: 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]
Thus, the final answer is:
[tex]\[
P(3) = 0.4
\][/tex]
