To determine the probability that a randomly selected student is either female or got an A, we need to find the probability of each event separately, as well as the probability that both events occur simultaneously. Here is the detailed step-by-step solution:
1. Total number of students:
The total number of students is given as 66.
2. Number of females:
The number of female students is given as 36.
3. Number of students who got an A:
The number of students who received an A is given as 7.
4. Number of females who got an A:
From the table, the number of females who got an A grade is 3.
Now, we calculate the individual probabilities:
5. Probability of selecting a female:
[tex]\[
P(\text{female}) = \frac{\text{Number of females}}{\text{Total number of students}} = \frac{36}{66}
\][/tex]
6. Probability of selecting a student who got an A:
[tex]\[
P(\text{A}) = \frac{\text{Number of students who got an A}}{\text{Total number of students}} = \frac{7}{66}
\][/tex]
7. Probability of selecting a female who got an A:
[tex]\[
P(\text{female and A}) = \frac{\text{Number of females who got an A}}{\text{Total number of students}} = \frac{3}{66}
\][/tex]
Using these probabilities, we can apply the formula for the union of two events:
8. Formula for the probability of the union of two events:
[tex]\[
P(\text{female or A}) = P(\text{female}) + P(\text{A}) - P(\text{female and A})
\][/tex]
Substitute the values into the formula:
9. Calculate the probability:
[tex]\[
P(\text{female or A}) = \frac{36}{66} + \frac{7}{66} - \frac{3}{66}
\][/tex]
Combine the fractions:
[tex]\[
P(\text{female or A}) = \frac{36 + 7 - 3}{66} = \frac{40}{66}
\][/tex]
10. Reduce the fraction:
To simplify [tex]\(\frac{40}{66}\)[/tex], we find the greatest common divisor (GCD) of 40 and 66, which is 2. Therefore:
[tex]\[
\frac{40}{66} = \frac{40 \div 2}{66 \div 2} = \frac{20}{33}
\][/tex]
So, the probability that a randomly selected student is either female or got an A is:
[tex]\[
P(\text{female or "A"}) = \frac{20}{33}
\][/tex]
Thus, the probability is [tex]\(\boxed{\frac{20}{33}}\)[/tex].
