Suppose a high school has partnerships with local businesses that allows juniors and seniors to participate in internships during the school year. During a specific year 
30
%
30%​​ of juniors and seniors intern at local businesses, 
65
%
65%​​ of juniors and seniors are involved in extracurricular activities, and 
24.5
%
24.5%​​ of juniors and seniors participate in internships and extracurricular activities.
What is the probability a junior or senior has an internship given that they participate in an extracurricular activity?What is the probability a junior or senior participates in an extracurricular activity given that they have an internship?

What is the probability that a juni or or senior has an internship or participates in an extracurricular
activity?
Are the events of having an internship and participating in an extracurricular activity independent?ALL ANSWERS AS FRACTIONS

Let's define the events:

- \( I \): A junior or senior has an internship.
- \( E \): A junior or senior participates in an extracurricular activity.

We are given the following probabilities:

- \( P(I) = 0.30 \)
- \( P(E) = 0.65 \)
- \( P(I \cap E) = 0.245 \)

### Probability that a junior or senior has an internship given that they participate in an extracurricular activity

We need to find \( P(I \mid E) \):

\[
P(I \mid E) = \frac{P(I \cap E)}{P(E)}
\]

Substituting the given values:

\[
P(I \mid E) = \frac{0.245}{0.65}
\]

To convert this to a fraction:

\[
P(I \mid E) = \frac{245}{650} = \frac{49}{130}
\]

Thus, \( P(I \mid E) = \boxed{\frac{49}{130}} \).

### Probability that a junior or senior participates in an extracurricular activity given that they have an internship

We need to find \( P(E \mid I) \):

\[
P(E \mid I) = \frac{P(I \cap E)}{P(I)}
\]

Substituting the given values:

\[
P(E \mid I) = \frac{0.245}{0.30}
\]

To convert this to a fraction:

\[
P(E \mid I) = \frac{245}{300} = \frac{49}{60}
\]

Thus, \( P(E \mid I) = \boxed{\frac{49}{60}} \).

### Probability that a junior or senior has an internship or participates in an extracurricular activity

We need to find \( P(I \cup E) \):

\[
P(I \cup E) = P(I) + P(E) - P(I \cap E)
\]

Substituting the given values:

\[
P(I \cup E) = 0.30 + 0.65 - 0.245
\]

\[
P(I \cup E) = 0.95 - 0.245
\]

\[
P(I \cup E) = 0.705
\]

To convert this to a fraction:

\[
P(I \cup E) = \frac{705}{1000} = \frac{141}{200}
\]

Thus, \( P(I \cup E) = \boxed{\frac{141}{200}} \).

### Are the events of having an internship and participating in an extracurricular activity independent?

Two events \( I \) and \( E \) are independent if \( P(I \cap E) = P(I) \cdot P(E) \). Let's check if this is true:

\[
P(I) \cdot P(E) = 0.30 \cdot 0.65 = 0.195
\]

We are given \( P(I \cap E) = 0.245 \).

Since \( 0.195 \neq 0.245 \), the events \( I \) and \( E \) are not independent.

Do you want details or have any questions?

Here are 5 questions you might ask:
1. How do you determine if two events are independent?
2. What is the difference between conditional probability and joint probability?
3. Can you explain the steps to convert a decimal to a fraction?
4. How do you use the principle of inclusion-exclusion in probability?
5. What are the applications of conditional probability in real-life scenarios?

**Tip:** When determining whether events are independent, always verify that the product of their individual probabilities equals the joint probability.

Explore probability problems involving internships and extracurricular activities for juniors and seniors. Calculate conditional probabilities and assess the independence of events based on given data.

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