To solve this question, we break it down into two parts as specified:
### Part 1: Determine the Probability a Student is in Sports Given That They Are a Senior
This requires us to determine [tex]\( P(\text{Sports} \mid \text{Senior}) \)[/tex].
From the two-way table, observe the following:
- The total number of seniors is 35.
- The total number of seniors who are in sports is 25.
We apply the conditional probability formula:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{\text{Number of seniors in sports}}{\text{Total number of seniors}} \][/tex]
By using the numbers directly from the table:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{25}{35} \][/tex]
In simple mathematical terms:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{25}{35} = \frac{5}{7} \approx 0.714 \][/tex]
### Part 2: Determine the Probability That It's a Senior and in Sports
This requires us to determine [tex]\( P(\text{Senior and Sports}) \)[/tex].
From the provided data:
- The total number of students is 100.
- The number of seniors who are in sports is 25.
We apply the probability formula for combined events:
[tex]\[ P(\text{Senior and Sports}) = \frac{\text{Number of seniors in sports}}{\text{Total number of students}} \][/tex]
By using the numbers directly from the table:
[tex]\[ P(\text{Senior and Sports}) = \frac{25}{100} \][/tex]
Therefore:
[tex]\[ P(\text{Senior and Sports}) = \frac{25}{100} = \frac{1}{4} = 0.25 \][/tex]
### Summary of Results
1. The probability a student is in sports, given that they are a senior [tex]\( P(\text{Sports} \mid \text{Senior}) \)[/tex] is approximately 0.714.
2. The probability that it’s a senior in sports [tex]\( P(\text{Senior and Sports}) \)[/tex] is 0.25.
So, specifically for the given question:
[tex]\[ P(\text{Senior}) = 0.35 \][/tex]
[tex]\[ P(\text{Senior and Sports}) = 0.25 \][/tex]
These are the detailed solutions for the given probabilities.
