Question

Fifty students in the third-grade class listed their hair and eye colors in the table below: Brown hair Blonde hair Total Blue eyes 14 8 22 Brown eyes 16 12 28 30 20 50 Are the events "blonde hair" and "blue eyes" independent?
A) Yes, P(blonde hair) • P(blue eyes) = P(blonde hair ∩ blue eyes)
B) Yes, P(blonde hair) • P(blue eyes) ≠ P(blonde hair ∩ blue eyes)
C) No, P(blonde hair) • P(blue eyes) = P(blonde hair ∩ blue eyes)
D) No, P(blonde hair) • P(blue eyes) ≠ P(blonde hair ∩ blue eyes)

Answer 1

If two events A and B are independent, then P(A)*P(B) = P(A∩B).  
P(A) is the probability of being blonde.
P(B) is the probability of having blue eyes.
P(A∩B) is the probability of being blonde with blue eyes.

P(A)*P(B) = (20/50)*(22/50) = 440/2500 = 22/125.
P(A∩B) = 8/50.
These two probabilities are not equivalent; 22/125=0.176, while 8/50=0.16.  Our answer is D.