To determine the probability that a pitcher will throw exactly 2 pitches in an at-bat, follow these steps:
1. Calculate the Total Frequency:
Summing up the frequencies given for different pitches will give the total number of at-bats observed.
[tex]\[
\text{Total Frequency} = 12 \text{ (for 1 pitch)} + 16 \text{ (for 2 pitches)} + 32 \text{ (for 3 pitches)} + 12 \text{ (for 4 pitches)} + 8 \text{ (for 5 pitches)}
\][/tex]
Adding these values together:
[tex]\[
\text{Total Frequency} = 12 + 16 + 32 + 12 + 8 = 80
\][/tex]
2. Identify the Frequency of 2 Pitches:
According to the data, the frequency of exactly 2 pitches is:
[tex]\[
\text{Frequency of 2 Pitches} = 16
\][/tex]
3. Calculate the Probability:
Probability is calculated by dividing the frequency of the event of interest (2 pitches) by the total frequencies (total number of at-bats).
[tex]\[
P(2) = \frac{\text{Frequency of 2 Pitches}}{\text{Total Frequency}} = \frac{16}{80}
\][/tex]
4. Simplify the Fraction:
Simplify [tex]\(\frac{16}{80}\)[/tex] to its lowest term:
[tex]\[
\frac{16}{80} = \frac{16 \div 16}{80 \div 16} = \frac{1}{5}
\][/tex]
5. Convert to Decimal:
If needed, convert the fraction to a decimal for better understanding:
[tex]\[
\frac{1}{5} = 0.2
\][/tex]
Thus, the probability that a pitcher will throw exactly 2 pitches in an at-bat is:
[tex]\[
P(2) = 0.2
\][/tex]
To summarize, the probability [tex]\(P(2)\)[/tex] that a pitcher will throw exactly 2 pitches in an at-bat is
[tex]\[
\boxed{0.2}
\][/tex]
