Suppose data is gathered about students at a particular high school at the end of the year. 
30
%
30%​​ of the students have their driver’s license, 
75
%
75%​​ of the students have at least a 
3.0
3.0​ GPA, and 
22.5
%
22.5%​​ of the students have their driver’s license and at least a 
3.0
3.0​ GPA.
What is the probability that a student has their driver’s license or has at least a 
3.0
3.0​ GPA? as a fraction

To find the probability that a student has their driver’s license or has at least a 3.0 GPA, we can use the principle of inclusion-exclusion for probabilities. Let's define the events:

- \( D \): A student has their driver's license.
- \( G \): A student has at least a 3.0 GPA.

We are given the following probabilities:

- \( P(D) = 0.30 \)
- \( P(G) = 0.75 \)
- \( P(D \cap G) = 0.225 \)

We want to find \( P(D \cup G) \), the probability that a student has their driver's license or has at least a 3.0 GPA. According to the principle of inclusion-exclusion:

\[
P(D \cup G) = P(D) + P(G) - P(D \cap G)
\]

Substituting the given values:

\[
P(D \cup G) = 0.30 + 0.75 - 0.225
\]

\[
P(D \cup G) = 1.05 - 0.225
\]

\[
P(D \cup G) = 0.825
\]

Therefore, the probability that a student has their driver’s license or has at least a 3.0 GPA is 0.825. To express this as a fraction, we note that 0.825 is equivalent to:

\[
0.825 = \frac{825}{1000} = \frac{33}{40}
\]

Thus, the probability that a student has their driver’s license or has at least a 3.0 GPA is:

\[
\boxed{\frac{33}{40}}
\]

Do you want details or have any questions?

Here are 5 questions you might ask:
1. How do you calculate probabilities using the principle of inclusion-exclusion?
2. Can you explain how to convert a decimal to a fraction?
3. What is the principle of inclusion-exclusion in probability?
4. How do you calculate joint probabilities?
5. How do you simplify fractions?

**Tip:** When working with probabilities, always ensure that the events are clearly defined and the given probabilities are correctly interpreted to avoid errors in calculations.

Explore how to calculate the probability that a student at a high school has either a driver's license or at least a 3.0 GPA. Detailed explanation using the principle of inclusion-exclusion. Suitable for advanced high school students.

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