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?

Answer 1

No, the events "brown hair" and "brown eyes" are not independent.

Explanation:

The table that represents the hair and eye colors of thirty students of the fifth grade are given in table as:

Brown hair Blonde hair Total

Green eyes 9 6 15

Brown eyes 10 5 15

Total 19 11 30

No, the events "brown hair" and "brown eyes" are not independent.

Since, two events A and B are said to be independent if:

P(A∩B)=P(A)×P(B)

where P denotes the probability of an event.

Here we have:

A= students having brown hair.

B= students having brown eyes.

A∩B= students having both brown hair and brown eyes.

Now,

P(A)=19/30 (ratio of addition of first column to the total entries)

P(B)=15/30 ( ratio of addition of second row to the total entries)

Also,

P(A∩B)=10/30

Now as:

P(A∩B) ≠ P(A)×P(B)

Hence, the two events are not independent.