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!

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.