Question

Thirty students in the fifth grade class listed their hair and eye colors in the table below:

Brown hair Blonde hair Total
Green eyes 9 6 15
Brown eyes 10 5 15
19 11 30
Are the events "brown hair" and "brown eyes" independent? (1 point)

a
Yes, P(brown hair) ⋅ P(brown eyes) = P(brown hair ∩ brown eyes)

b
Yes, P(brown hair) ⋅ P(brown eyes) ≠ P(brown hair ∩ brown eyes)

c
No, P(brown hair) ⋅ P(brown eyes) = P(brown hair ∩ brown eyes)

d
No, P(brown hair) ⋅ P(brown eyes) ≠ P(brown hair ∩ brown eyes)

Answer 1

In general, if two events A and B are independent

P(A ∩ B) = P (A) * P(B)

otherwise they are not independent.

So in this case, you would calculate P(brown hair ∩ brown eyes) and P(brown hair) * P(brown eyes) and determine whether both sides are equal or not.

To calculate P(brown hair ∩ brown eyes), simply look at the table, find out how many people have *both* brown eyes and brown hair, then divide by the total # of people. You can leave it as a fraction.

To calculate P(brown hair), look at out many people have brown hair in total, and divide by the total number of people. Leave it as a fraction. Repeat the same logic for P(brown eyes).

Finally, set up P(brown hair) ⋅ P(brown eyes) = P(brown hair ∩ brown eyes and see if both sides of the equation are equal or not.