Let's solve the problem step-by-step.
The goal is to find the probability that a randomly selected person from the survey is 8 to 12 years old, given that their favorite sport is baseball. This can be represented as:
[tex]\[ P (8-12 \text{ yrs} \mid \text{Baseball}) \][/tex]
### Step 1: Identify the Relevant Data
- The number of people aged 8 to 12 years old who favor baseball: 10
- The total number of people who favor baseball: 46
### Step 2: Setup the Conditional Probability Formula
The formula for conditional probability, \( P(A \mid B) \), is given by:
[tex]\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Here:
- \( A \) is the event that the person is aged 8 to 12 years.
- \( B \) is the event that the person's favorite sport is baseball.
This transforms the formula to:
[tex]\[ P(8-12 \text{ yrs} \mid \text{Baseball}) = \frac{\text{Number of people aged 8-12 years who like baseball}}{\text{Total number of people who like baseball}} \][/tex]
Substituting the given values:
[tex]\[ P(8-12 \text{ yrs} \mid \text{Baseball}) = \frac{10}{46} \][/tex]
### Step 3: Calculate the Probability
To find the probability as a percentage, multiply the fraction by 100:
[tex]\[ P(8-12 \text{ yrs} \mid \text{Baseball}) = \left(\frac{10}{46}\right) \times 100 \approx 21.7391\% \][/tex]
### Step 4: Round the Answer to the Nearest Whole Percent
Rounding 21.7391 to the nearest whole percent gives us:
[tex]\[ P(8-12 \text{ yrs} \mid \text{Baseball}) \approx 22\% \][/tex]
Therefore, the probability that a randomly selected person from this survey is 8 to 12 years old, given their favorite sport is baseball is approximately [tex]\( 22\% \)[/tex].
