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Giving a test to a group of students, the grades and gender are summarized below:

\begin{tabular}{|r|r|r|r|r|}
\hline & A & B & C & Total \\
\hline Male & 11 & 13 & 6 & 30 \\
\hline Female & 4 & 9 & 5 & 18 \\
\hline Total & 15 & 22 & 11 & 48 \\
\hline
\end{tabular}

If one student is chosen at random:

1. Find the probability that the student got an [tex]$A$[/tex]: [tex]$\square$[/tex]
2. Find the probability that the student was female [tex]$AND$[/tex] got an [tex]$A$[/tex]: [tex]$\square$[/tex]
3. Find the probability that the student was female [tex]$OR$[/tex] got a [tex]$C$[/tex]: [tex]$\square$[/tex]
4. Find the probability that the student was female [tex]$GIVEN$[/tex] they got a [tex]$B$[/tex]: [tex]$\square$[/tex]

Sagot :

GinnyAnsweridn
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]
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