Let's break down the problem step by step.

Given Data:

• Total number of students: 49
• Number of men: 11
• Number of women: 38
• Number of men who received an A: 2
• Number of women who received an A: 3

Part (a): Find the probability that the student is a woman.

The probability $P(\text{woman})$ that the student is a woman is calculated as:

$P(\text{woman}) = \frac{\text{Number of women}}{\text{Total number of students}} = \frac{38}{49}$

Part (b): Find the probability that the student received an A.

First, find the total number of students who received an A:

$\text{Total number of A's} = \text{Number of men who received an A} + \text{Number of women who received an A} = 2 + 3 = 5$

The probability $P(A)$ that the student received an A is:

$P(A) = \frac{\text{Total number of A's}}{\text{Total number of students}} = \frac{5}{49}$

Part (c): Find the probability that the student is a woman or received an A.

To find this, we use the formula for the union of two events:

$P(\text{woman or A}) = P(\text{woman}) + P(A) - P(\text{woman and A})$

We already calculated $P(\text{woman}) = \frac{38}{49}$ and $P(A) = \frac{5}{49}$.

Now, we calculate the probability $P(\text{woman and A})$, which is the probability that the student is both a woman and received an A:

$P(\text{woman and A}) = \frac{\text{Number of women who received an A}}{\text{Total number of students}} = \frac{3}{49}$

Now, calculate $P(\text{woman or A})$:

$P(\text{woman or A}) = \frac{38}{49} + \frac{5}{49} - \frac{3}{49} = \frac{40}{49}$

Part (d): Find the probability that the student did not receive an A.

The probability that the student did not receive an A is the complement of the probability that the student received an A:

$P(\text{not A}) = 1 - P(A) = 1 - \frac{5}{49} = \frac{44}{49}$

Final Answers:

(a) The probability that the student is a woman: $\frac{38}{49}$

(b) The probability that the student received an A: $\frac{5}{49}$

(c) The probability that the student is a woman or received an A: $\frac{40}{49}$

(d) The probability that the student did not receive an A: $\frac{44}{49}$

Would you like more details on any of these calculations, or do you have any questions?

Here are 5 related questions you might find interesting:

1. What is the probability that the student is a man?
2. If a student received an A, what is the probability that they are a woman?
3. What is the probability that a randomly chosen man did not receive an A?
4. If you know a student is a woman, what is the probability that she received an A?
5. What is the probability that a randomly chosen student is a man and did not receive an A?

Tip: Remember, the sum of the probabilities of complementary events (like receiving an A and not receiving an A) is always 1.