Question

Statistics grades: In a statistics class of 41 students, there were 19 men and 22 women. Three of the men and four 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.

Answer 1

To solve the given problems, we will first summarize the given information:

- Total number of students = 41
- Number of men = 19
- Number of women = 22
- Men who received an A = 3
- Women who received an A = 4
- Total students who received an A = 3 (men) + 4 (women) = 7

Now, we will calculate each probability step-by-step.

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

The probability that the student is a woman is given by the number of women divided by the total number of students:

\[
P(\text{Woman}) = \frac{\text{Number of women}}{\text{Total number of students}} = \frac{22}{41}
\]

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

The probability that the student received an A is given by the number of students who received an A divided by the total number of students:

\[
P(\text{A}) = \frac{\text{Number of students who received an A}}{\text{Total number of students}} = \frac{7}{41}
\]

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

To find the probability that the student is a woman or received an A, we can use the principle of inclusion-exclusion:

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

We already know:
- \(P(\text{Woman}) = \frac{22}{41}\)
- \(P(\text{A}) = \frac{7}{41}\)

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

\[
P(\text{Woman and A}) = \frac{\text{Number of women who received an A}}{\text{Total number of students}} = \frac{4}{41}
\]

Now we can substitute these values:

\[
P(\text{Woman or A}) = \frac{22}{41} + \frac{7}{41} - \frac{4}{41} = \frac{22 + 7 - 4}{41} = \frac{25}{41}
\]

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

The probability that the student did not receive an A can be found by subtracting the probability of receiving an A from 1:

\[
P(\text{Not A}) = 1 - P(\text{A}) = 1 - \frac{7}{41} = \frac{41 - 7}{41} = \frac{34}{41}
\]

### Summary of results:

(a) \(P(\text{Woman}) = \frac{22}{41}\)
(b) \(P(\text{A}) = \frac{7}{41}\)
(c) \(P(\text{Woman or A}) = \frac{25}{41}\)
(d) \(P(\text{Not A}) = \frac{34}{41}\)