From Kenesha’s table, you know that 44 students surveyed were boys, 20 students preferred basketball, and 14 students were boys who prefer basketball. Find the conditional relative frequency that a student surveyed is a boy, given that the student prefers basketball. Express your answer as a decimal and as a percent.
The final answer given here is correct, but the intermediate steps are not. There is a piece of information that you have not given us, namely the total number of students surveyed.
Suppose an equal number of boys and girls were surveyed, so there were 44 girls for a total of 88 students.
Then P(A and B)=14/88 and P(B)=20/88. And (14/88)/(20/88)=14/20=.7 as in the answer. But it should be clear that the answer in this case does not depend on the total number of students surveyed, while in another question it might change the answer.
The 44 used in the question is the total number of boys -- certainly if 14 boys preferred basketball and 20 students overall preferred basketball, then at least 6 of the students are not boys.
We can see that the answer is correct-- 20 students preferred basketball, and of those 14 are boys. So the probability that a student is male, given that they prefer basketball, is 14/20.
Take note of the formula for the conditional probability:
`P(A|B) = (P(A and B))/(P(B))`
Set A = event that the student surveyed is a boy.
B = event that the student surveyed prefers basketball.
Therefore, `P(A and B) = 14/44`
`P(B) = 20/44` .
Applying the formula:
`P(A|B) = (14/44)/(20/44) `
In dividing fractions, we flip the bottom and proceed to multiplication.
`P(A|B) = (14/44)/(20/44) = (14/44) * (44/20)`
Cancelling the common factors, we will have:
P(A|B) = 14/20 = 0.7 or 70%.
Therefore, the conditional relative frequency that a student surveyed is a boy, given that the student prefers basketball is 0.7 or 70%.