Sure, let's go through each part step by step based on the given data in the table.
1. Probability that the student got an [tex]$A$[/tex]:
To find this probability, we simply divide the number of students who got an A by the total number of students.
[tex]\[ \text{P(A)} = \frac{\text{Number of students who got an A}}{\text{Total number of students}} = \frac{15}{48} = 0.3125 \][/tex]
2. Probability that the student was female AND got an [tex]$A$[/tex]:
To find this probability, we need the number of females who got an A divided by the total number of students.
[tex]\[ \text{P(Female and A)} = \frac{\text{Number of females who got an A}}{\text{Total number of students}} = \frac{4}{48} = 0.0833 \][/tex]
3. Probability that the student was female OR got a [tex]$C$[/tex]:
To find this probability, we use the formula for the probability of the union of two events:
[tex]\[ \text{P(Female or C)} = \text{P(Female)} + \text{P(C)} - \text{P(Female and C)} \][/tex]
Where:
- [tex]\( \text{P(Female)} \)[/tex] is the probability of a student being female;
- [tex]\( \text{P(C)} \)[/tex] is the probability of a student getting a C;
- [tex]\( \text{P(Female and C)} \)[/tex] is the probability of a student being female and getting a C.
Calculating these individually:
[tex]\[ \text{P(Female)} = \frac{\text{Total number of females}}{\text{Total number of students}} = \frac{18}{48} \][/tex]
[tex]\[ \text{P(C)} = \frac{\text{Number of students who got a C}}{\text{Total number of students}} = \frac{11}{48} \][/tex]
[tex]\[ \text{P(Female and C)} = \frac{\text{Number of females who got a C}}{\text{Total number of students}} = \frac{5}{48} \][/tex]
Combining these:
[tex]\[ \text{P(Female or C)} = \left( \frac{18}{48} + \frac{11}{48} - \frac{5}{48} \right) = \frac{24}{48} = 0.5 \][/tex]
4. Probability that the student was female GIVEN they got a B:
To find this conditional probability, we divide the number of females who got a B by the total number of students who got a B:
[tex]\[ \text{P(Female | B)} = \frac{\text{Number of females who got a B}}{\text{Total number of students who got a B}} = \frac{9}{22} = 0.4091 \][/tex]
So the final probabilities are:
1. Probability that the student got an [tex]$A$[/tex]: [tex]\( \boxed{0.3125} \)[/tex]
2. Probability that the student was female AND got an [tex]$A$[/tex]: [tex]\( \boxed{0.0833} \)[/tex]
3. Probability that the student was female OR got a [tex]$C$[/tex]: [tex]\( \boxed{0.5} \)[/tex]
4. Probability that the student was female GIVEN they got a B: [tex]\( \boxed{0.4091} \)[/tex]
