To solve the problem of finding the probability that a randomly chosen student is either female or received a grade of "B," let's go through the necessary steps methodically.
### Step 1: Identify Total Number of Students
First, we note the total number of students:
[tex]\[ \text{Total students} = 64 \][/tex]
### Step 2: Calculate Number of Females
Next, we identify the total number of female students:
[tex]\[ \text{Total females} = 24 \][/tex]
### Step 3: Calculate Number of Grade "B" Students
We then count the total number of students who received a grade "B":
[tex]\[ \text{Total B grades} = 22 \][/tex]
### Step 4: Identify Students who are Both Female and Received a Grade "B"
We are also informed that there are 5 female students who scored a grade "B".
### Step 5: Calculate Individual Probabilities
We can now determine the individual probabilities:
1. Probability that a student is female:
[tex]\[ P(\text{Female}) = \frac{\text{Total females}}{\text{Total students}} = \frac{24}{64} = 0.375 \][/tex]
2. Probability that a student received a grade "B":
[tex]\[ P(\text{Grade B}) = \frac{\text{Total B grades}}{\text{Total students}} = \frac{22}{64} = 0.3438 \][/tex]
3. Probability that a student is both female and received a grade "B":
[tex]\[ P(\text{Female and Grade B}) = \frac{\text{Female and B grade}}{\text{Total students}} = \frac{5}{64} = 0.0781 \][/tex]
### Step 6: Use the Inclusion-Exclusion Principle
To find the probability that a student is either female or received a grade "B", we apply the inclusion-exclusion principle:
[tex]\[ P(\text{Female or Grade B}) = P(\text{Female}) + P(\text{Grade B}) - P(\text{Female and Grade B}) \][/tex]
Substitute the calculated probabilities into the formula:
[tex]\[ P(\text{Female or Grade B}) = 0.375 + 0.3438 - 0.0781 = 0.6406 \][/tex]
Thus, the probability that a randomly chosen student is either female or received a grade "B" is:
[tex]\[ \boxed{0.6406} \][/tex]
