To solve this problem, we can proceed step-by-step to find the probability that a randomly chosen student is either male or received a grade of "A". This involves calculating several intermediate probabilities and then combining them using the principle of inclusion-exclusion.
Here is the detailed solution:
1. Total Number of Students:
The total number of students is the sum of all male and female students:
[tex]\[
\text{Total Students} = 47 + 24 = 71
\][/tex]
2. Total Number of Males:
According to the table, the total number of male students is:
[tex]\[
\text{Total Males} = 47
\][/tex]
3. Total Number of Students who Received an "A":
The students who received an "A" include both males and females who received an "A":
[tex]\[
\text{Total A} = 14 (\text{Males who received an A}) + 17 (\text{Females who received an A}) = 31
\][/tex]
4. Number of Male Students who Received an "A":
We already have this information from the table:
[tex]\[
\text{Male A} = 14
\][/tex]
5. Calculate the Probability of Selecting a Male or a Student who Received an "A":
Using the principle of inclusion-exclusion for probability:
[tex]\[
P(\text{Male OR A}) = P(\text{Male}) + P(\text{A}) - P(\text{Male AND A})
\][/tex]
Where:
[tex]\[
P(\text{Male}) = \frac{\text{Total Males}}{\text{Total Students}} = \frac{47}{71}
\][/tex]
[tex]\[
P(\text{A}) = \frac{\text{Total A}}{\text{Total Students}} = \frac{31}{71}
\][/tex]
[tex]\[
P(\text{Male AND A}) = \frac{\text{Male A}}{\text{Total Students}} = \frac{14}{71}
\][/tex]
6. Substitute the Values into the Formula:
[tex]\[
P(\text{Male OR A}) = \frac{47}{71} + \frac{31}{71} - \frac{14}{71}
\][/tex]
[tex]\[
P(\text{Male OR A}) = \frac{47 + 31 - 14}{71} = \frac{64}{71}
\][/tex]
7. Convert the Fraction to a Decimal:
[tex]\[
P(\text{Male OR A}) \approx 0.9014
\][/tex]
Therefore, the probability that a randomly chosen student is either male or received a grade of "A" is [tex]\(0.9014\)[/tex] when rounded to four decimal places.
