**Statistics grades:** In a statistics class of 

49

 students, there were 

11

 men and 

38

 women. Two of the men and three of the women received an A in the course. A student is chosen at random from the class.

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

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

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

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

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.

Learn how to calculate probabilities in a statistics class scenario involving students, genders, and grades. Explore the probabilities of students being women, receiving an A, both, or neither. Suitable for high school students studying probability in statistics.

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