Question

Suppose that among the 6,000 students at a high school, 1,500 are taking honors courses and 1,800 prefer watching basketball to watching football. If 450 students are both taking honors courses and prefer basketball to football, are the events "taking honors courses" and "preferring basketball" independent? No, because StartFraction 1500 Over 6000 EndFraction not-equals StartFraction 1800 Over 6000 EndFraction Yes, because StartFraction 450 Over 1500 EndFraction not-equals StartFraction 450 Over 1800 EndFraction No, because (1,500)(1,800) ≠ 450. Yes, because (StartFraction 1500 Over 6000 EndFraction) (StartFraction 1800 Over 6000 EndFraction) = StartFraction 450 Over 6000 EndFraction.

Answer 1

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

`A\cap B` : Both

We know that,

`\text{Probability}=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}`

Using this formula, we get

`P(A)=(1500)/(6000)=0.25`

`P(B)=(1800)/(6000)=0.30`

`P(A\cap B)=(450)/(6000)=0.075`

Two evens are independent if

`P(E_1\cap E_2)P(E_1)\cdot P(E_2)`

Now,

`P(A)\cdot P(B)=0.25* 0.30`

`P(A)\cdot P(B)=0.075`

`P(A)\cdot P(B)=P(A\cap B)`

So, the events A and B are intendent because

`((1500)/(6000))((1800)/(6000))=(450)/(6000)`

Therefore, the correct option is D.