[tex]\begin{gathered} 1)\text{ }\frac{28}{45} \\ \\ 2)\text{ }\frac{8}{45} \\ \\ 3)\text{ }\frac{40}{65} \end{gathered}[/tex]
Here, we want to calculate probabilities;
We have this as follows;
1) We want to calculate the probability that a randomly selected student is a 9th grader or a student that preferred unit 7
From here, we need the number of students who are 9th graders and students that prefer unit 7
From the question, we have it that 1/3 of the total students are 9th graders
So, for a total of 450, the number of 9th graders will be 1/3 * 450 = 150 students
Secondly we need the number of students that prefers unit 7
Let us try and complete the table as follows;
From the completed table, the numbers that like unit 7 are 130
So the probability we want to calculate is the sum of the two divided by 450
We have this as;
[tex]\frac{130+150}{450}\text{ = }\frac{280}{450}\text{ = }\frac{28}{45}[/tex]
2) Here, we want to calculate the probability that a randomly selected student is a 10th grader who also prefers unit 8
From the table, we can see that the number of students who are 10th graders and also prefer unit 8 is 80
So, we have the probability as;
[tex]\frac{80}{450}\text{ = }\frac{8}{45}[/tex]
3) Here, we want to calculate the probability that given that a student prefers unit 5, what is the probability that he is a 10th grader
We use the conditional probability value here
Where event A is the probability that student is a 10th grader, while event B is the probability that a student prefers unit 5
We have the probability as;
[tex]\begin{gathered} P(A|B)\text{ = }\frac{P(AnB)}{P(B)} \\ \\ P(\text{AnB) = }\frac{40}{450};\text{ P(B) = }\frac{65}{450} \\ \\ P(A|B)\text{ = }\frac{40}{65} \end{gathered}[/tex]
