Giving a test to a group of students, the grades and gender are summarized below



A

B

C

Total

Male

20

14

6

40

Female

4

15

10

29

Total

24

29

16

69


If one student is chosen at random,

Find the probability that the student got a C:   

Find the probability that the student was female AND got a "A":   

Find the probability that the student was male OR got an "A":   

If one student is chosen at random, find the probability that the student got a 'C' GIVEN they are male:   

We can solve these probability questions step-by-step based on the provided table. The total number of students is \(69\), as indicated by the sum of all entries in the table. 

### Table Summary:
|      | A   | B   | C   | Total |
|------|-----|-----|-----|-------|
| Male | 20  | 14  | 6   | 40    |
| Female | 4   | 15  | 10  | 29    |
| **Total**  | 24  | 29  | 16  | 69    |

### 1. Find the probability that the student got a C:

To find this probability, we need the total number of students who got a "C" divided by the total number of students.

\[
P(\text{C}) = \frac{\text{Total students who got a C}}{\text{Total students}} = \frac{16}{69}
\]

So, the probability is:

\[
P(\text{C}) = \frac{16}{69}
\]

### 2. Find the probability that the student was female AND got an "A":

For this, we need to count how many females got an "A" and divide by the total number of students.

\[
P(\text{Female and A}) = \frac{\text{Females who got an A}}{\text{Total students}} = \frac{4}{69}
\]

So, the probability is:

\[
P(\text{Female and A}) = \frac{4}{69}
\]

### 3. Find the probability that the student was male OR got an "A":

This is a union probability \(P(\text{Male} \cup \text{A})\), which can be found using the formula:

\[
P(\text{Male or A}) = P(\text{Male}) + P(\text{A}) - P(\text{Male and A})
\]

Where:
- \(P(\text{Male}) = \frac{40}{69}\) (since there are 40 males)
- \(P(\text{A}) = \frac{24}{69}\) (since 24 students got an "A")
- \(P(\text{Male and A}) = \frac{20}{69}\) (since 20 males got an "A")

Now, applying the formula:

\[
P(\text{Male or A}) = \frac{40}{69} + \frac{24}{69} - \frac{20}{69} = \frac{44}{69}
\]

So, the probability is:

\[
P(\text{Male or A}) = \frac{44}{69}
\]

### 4. Find the probability that the student got a 'C' GIVEN they are male:

This is a conditional probability \(P(\text{C}|\text{Male})\), which is given by:

\[
P(\text{C | Male}) = \frac{P(\text{Male and C})}{P(\text{Male})}
\]

Where:
- \(P(\text{Male and C}) = \frac{6}{69}\) (since 6 males got a "C")
- \(P(\text{Male}) = \frac{40}{69}\)

Now, the conditional probability is:

\[
P(\text{C | Male}) = \frac{\frac{6}{69}}{\frac{40}{69}} = \frac{6}{40} = \frac{3}{20}
\]

So, the probability is:

\[
P(\text{C | Male}) = \frac{3}{20}
\]

### Summary of Answers:
1. \( P(\text{C}) = \frac{16}{69} \)
2. \( P(\text{Female and A}) = \frac{4}{69} \)
3. \( P(\text{Male or A}) = \frac{44}{69} \)
4. \( P(\text{C | Male}) = \frac{3}{20} \)

Would you like further details on any of these calculations or have any questions?

Here are five related questions:
1. What is the probability that the student was male AND got a "B"?
2. What is the probability that a randomly chosen student was female OR got a "C"?
3. If the student was female, what is the probability that she got a "B"?
4. What is the probability that a randomly chosen student got a "B"?
5. What is the probability that a randomly chosen student was female AND got a "C"?

**Tip:** Conditional probabilities focus on a specific subgroup, which makes them useful in many real-life scenarios, such as diagnosing medical conditions or targeted marketing.

Giving a test to a group of students, the grades and gender are summarized below: Male (A: 20, B: 14, C: 6, Total: 40), Female (A: 4, B: 15, C: 10, Total: 29), Total (A: 24, B: 29, C: 16, Total: 69). Find the probabilities for: 1) a student got a C, 2) a student was female AND got an A, 3) a student was male OR got an A, 4) a student got a C GIVEN they are male.

Learn how to calculate probabilities based on gender and grades distribution, including conditional and union probabilities. This problem explores key probability concepts suitable for students in Grades 9-12. Includes detailed step-by-step solutions for finding the probability of getting a C, being female and getting an A, and more.

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