Let’s solve each part of this problem step-by-step.

Given Information:

• Total students = 39
• Men = 10, Women = 29
• Students who received an A: 3 men and 4 women

Solution:

(a) Probability that the student is a woman

The probability $P(\text{Woman})$ is calculated by dividing the number of women by the total number of students.

$P(\text{Woman}) = \frac{\text{Number of Women}}{\text{Total Students}} = \frac{29}{39}$

Calculating this value:

$P(\text{Woman}) \approx 0.7436$

(b) Probability that the student received an A

The probability $P(\text{Received an A})$ is calculated by dividing the total number of students who received an A by the total number of students.

Total students who received an A = 3 (men) + 4 (women) = 7

$P(\text{Received an A}) = \frac{7}{39}$

Calculating this value:

$P(\text{Received an A}) \approx 0.1795$

(c) Probability that the student is a woman or received an A

To find $P(\text{Woman or Received an A})$, we use the formula for the union of two events:

$P(\text{Woman or Received an A}) = P(\text{Woman}) + P(\text{Received an A}) - P(\text{Woman and Received an A})$

From above:

• $P(\text{Woman}) = 0.7436$
• $P(\text{Received an A}) = 0.1795$

Now, $P(\text{Woman and Received an A})$ is the probability that a student is a woman and received an A, which is calculated by dividing the number of women who received an A by the total number of students.

$P(\text{Woman and Received an A}) = \frac{4}{39} \approx 0.1026$

Substituting these values:

$P(\text{Woman or Received an A}) = 0.7436 + 0.1795 - 0.1026 \approx 0.8205$

(d) Probability that the student did not receive an A

The probability $P(\text{Did not receive an A})$ is the complement of the probability of receiving an A.

$P(\text{Did not receive an A}) = 1 - P(\text{Received an A})$

From (b), we know $P(\text{Received an A}) \approx 0.1795$, so:

$P(\text{Did not receive an A}) = 1 - 0.1795 = 0.8205$

Summary of Answers:

1. (a) Probability that the student is a woman: $\approx 0.7436$
2. (b) Probability that the student received an A: $\approx 0.1795$
3. (c) Probability that the student is a woman or received an A: $\approx 0.8205$
4. (d) Probability that the student did not receive an A: $\approx 0.8205$

Would you like a detailed explanation of any step or have any questions?

Related Questions

1. What is the probability that a student chosen is a man?
2. What is the probability that the student is a man and received an A?
3. How would the probability change if there were 5 women who received an A?
4. What is the probability that a randomly chosen student is a woman who did not receive an A?
5. How would you calculate the probability if there was a different total number of students?

Tip

Always verify if events are mutually exclusive or independent, as this affects which probability rules to apply.