After giving a test to a group of students, below is the summary of grades by gender.
A	B	C	Total
Male	19	2	17	38
Female	5	11	15	31
Total	24	13	32	69
If a student is randomly chosen, find the probability that the student is female or earned grade A. Leave your answer as a reduced fraction.

To find the probability that a randomly chosen student is either female or earned grade A, we'll use the principle of inclusion-exclusion.

### Step 1: Identify the relevant values.
- Total number of students: \( 69 \)
- Number of females: \( 31 \)
- Number of students who earned grade A: \( 24 \)
- Number of female students who earned grade A: \( 5 \)

### Step 2: Calculate the individual probabilities.
- Probability of selecting a female student: 
  \[
  P(\text{Female}) = \frac{31}{69}
  \]
- Probability of selecting a student with grade A:
  \[
  P(\text{Grade A}) = \frac{24}{69}
  \]

### Step 3: Calculate the probability of selecting a female student who earned grade A.
This value was given as 5 out of 69:
\[
P(\text{Female and Grade A}) = \frac{5}{69}
\]

### Step 4: Apply the principle of inclusion-exclusion.
To find the probability of selecting a student who is either female or earned grade A, we add the individual probabilities and subtract the overlap:
\[
P(\text{Female or Grade A}) = P(\text{Female}) + P(\text{Grade A}) - P(\text{Female and Grade A})
\]
Substituting the values:
\[
P(\text{Female or Grade A}) = \frac{31}{69} + \frac{24}{69} - \frac{5}{69} = \frac{31 + 24 - 5}{69} = \frac{50}{69}
\]

### Final Answer:
The probability that a randomly chosen student is either female or earned grade A is \(\frac{50}{69}\).

Would you like more details or have any questions?

Here are five related questions:
1. What is the probability of selecting a male student who earned grade B?
2. If a student is chosen randomly, what is the probability that the student earned grade C?
3. What is the probability that a student selected is male or earned grade B?
4. How do you calculate the probability of two events happening together?
5. What is the complement of the probability calculated above?

**Tip:** In probability problems, the principle of inclusion-exclusion helps avoid double-counting when dealing with "or" events.

Explore how to calculate the probability of selecting a female or a student who earned grade A in a group of 69 students. This problem involves the principle of inclusion-exclusion and basic probability concepts, suitable for Grades 8-10 students.

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