A test was given to a group of students. The grades and gender are summarized below

A

B

C

Total

Male

8

6

18

32

Female

19

4

7

30

Total

27

10

25

62

If one student is chosen at random from those who took the test,

Find the probability that the student was female GIVEN they got an 'A'.

To find the probability that a randomly chosen student was female given that they got an 'A', we use conditional probability.

The formula for conditional probability is:

\[
P(\text{Female} \mid A) = \frac{P(\text{Female} \cap A)}{P(A)}
\]

Where:
- \(P(\text{Female} \mid A)\) is the probability that the student is female given that they got an 'A'.
- \(P(\text{Female} \cap A)\) is the probability that the student is female and got an 'A'.
- \(P(A)\) is the probability that a student got an 'A'.

### Step 1: Find \(P(\text{Female} \cap A)\)
From the table, we see that 19 females got an 'A'. The total number of students is 62. So,

\[
P(\text{Female} \cap A) = \frac{19}{62}
\]

### Step 2: Find \(P(A)\)
The total number of students who got an 'A' (both males and females) is 27. Thus,

\[
P(A) = \frac{27}{62}
\]

### Step 3: Calculate \(P(\text{Female} \mid A)\)

\[
P(\text{Female} \mid A) = \frac{\frac{19}{62}}{\frac{27}{62}} = \frac{19}{27}
\]

Thus, the probability that a student was female given that they got an 'A' is:

\[
P(\text{Female} \mid A) = \frac{19}{27}
\]

Let me know if you need more details on this, or have any questions!

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Here are five related questions to think about:

1. What is the probability that a randomly chosen student was male given they got an 'A'?
2. What is the probability that a student got a 'B' regardless of gender?
3. What is the probability that a randomly chosen student was female given they got a 'C'?
4. What is the probability that a student chosen at random got an 'A' and is male?
5. How would the probability change if there were more students in the group?

**Tip**: In conditional probability, always make sure to clearly identify the condition you are working with and adjust your sample space accordingly.

A test was given to a group of students. The grades and gender are summarized below

A
B
C
Total
Male
8
6
18
32
Female
19
4
7
30
Total
27
10
25
62

If one student is chosen at random from those who took the test, find the probability that the student was female GIVEN they got an 'A'.

Discover how to find the probability that a student is female given they received an 'A' in a test. The problem explores conditional probability using the formula P(Female | A) = P(Female ∩ A) / P(A), with data on student grades and gender. Suitable for high school students (Grades 9-12).

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