Below, a two-way table is given for student activities:

\begin{tabular}{|c|c|c|c|c|}
\hline
& Sports & Drama & Work & Total \\
\hline
Sophomore & 20 & 7 & 3 & 30 \\
\hline
Junior & 20 & 13 & 2 & 35 \\
\hline
Senior & 25 & 5 & 5 & 35 \\
\hline
Total & 65 & 25 & 10 & 100 \\
\hline
\end{tabular}

Find the probability that a student is in sports, given that they are a senior.

Use the formula below to get your final answer:
[tex]\[
\begin{array}{c}
P(\text{senior}) = 0.35 \\
P(\text{senior and sports}) = 0.25 \\
P(\text{sports} \mid \text{senior}) = \frac{P(\text{sports and senior})}{P(\text{senior})} = [?] \%
\end{array}
\][/tex]

Round your answer to the nearest whole percent.
[tex]\[\square\][/tex]



Answer :

To determine the probability that a student participates in sports given that they are a senior, we'll use the concept of conditional probability. Let's break down the steps to solve this problem systematically:

1. Understand the problem: We need the probability that a student participates in sports given that they are a senior, which is denoted \(P(\text{sports} \mid \text{senior})\).

2. Define the probabilities given:
- The probability that a student is a senior, \(P(\text{senior})\).
- The probability that a student is both in sports and a senior, \(P(\text{sports and senior})\).

3. Extract the required values from the table:
- The total number of students is 100.
- The number of seniors is 35.
- The number of seniors who participate in sports is 25.

4. Calculate the individual probabilities:
- \(P(\text{senior})\) is calculated as the number of seniors divided by the total number of students:
[tex]\[ P(\text{senior}) = \frac{35}{100} = 0.35 \][/tex]

- \(P(\text{sports and senior})\) is calculated as the number of seniors who participate in sports divided by the total number of students:
[tex]\[ P(\text{sports and senior}) = \frac{25}{100} = 0.25 \][/tex]

5. Apply the formula for conditional probability:
The formula for conditional probability is given by:
[tex]\[ P(\text{sports} \mid \text{senior}) = \frac{P(\text{sports and senior})}{P(\text{senior})} \][/tex]
Plugging in the values we have:
[tex]\[ P(\text{sports} \mid \text{senior}) = \frac{0.25}{0.35} \][/tex]

6. Perform the division:
[tex]\[ P(\text{sports} \mid \text{senior}) = \frac{0.25}{0.35} \approx 0.7143 \][/tex]

7. Convert to a percentage and round to the nearest whole number:
[tex]\[ P(\text{sports} \mid \text{senior}) \times 100\% \approx 71.43\% \][/tex]
Rounded to the nearest whole number, this is 71%.

Answer: The probability that a student is in sports given that they are a senior is approximately 71%.