Given:
Total students = 6,000
Students taking honors courses = 1,500
Students preferring basketball = 1,800
Students in both = 450
To find:
Whether the two events "taking honors courses" and "preferring basketball" are independent or not?
Solution:
Let as consider the following events.
A : Taking honors courses
B : Preferring basketball
[tex]A\cap B[/tex] : Both
We know that,
[tex]\text{Probability}=\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}[/tex]
Using this formula, we get
[tex]P(A)=\dfrac{1500}{6000}=0.25[/tex]
[tex]P(B)=\dfrac{1800}{6000}=0.30[/tex]
[tex]P(A\cap B)=\dfrac{450}{6000}=0.075[/tex]
Two evens are independent if
[tex]P(E_1\cap E_2)P(E_1)\cdot P(E_2)[/tex]
Now,
[tex]P(A)\cdot P(B)=0.25\times 0.30[/tex]
[tex]P(A)\cdot P(B)=0.075[/tex]
[tex]P(A)\cdot P(B)=P(A\cap B)[/tex]
So, the events A and B are intendent because
[tex](\dfrac{1500}{6000})(\dfrac{1800}{6000})=\dfrac{450}{6000}[/tex]
Therefore, the correct option is D.
