Answer:
a) P(male=blue or female=blue) = 0.71
b) P(female=blue | male=blue) = 0.68
c) P(female=blue | male=brown) = 0.35
d) P(female=blue | male=green) = 0.31
e) We can conclude that the eye colors of male respondents and their partners are not independent.
Step-by-step explanation:
We are given following information about eye colors of 204 Scandinavian men and their female partners.
Blue Brown Green Total
Blue 78 23 13 114
Brown 19 23 12 54
Green 11 9 16 36
Total 108 55 41 204
a) What is the probability that a randomly chosen male respondent or his partner has blue eyes?
Using the addition rule of probability,
∵ P(A or B) = P(A) + P(B) - P(A and B)
For the given case,
P(male=blue or female=blue) = P(male=blue) + P(female=blue) - P(male=blue and female=blue)
P(male=blue or female=blue) = 114/204 + 108/204 − 78/204
P(male=blue or female=blue) = 0.71
b) What is the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes?
As per the rule of conditional probability,
P(female=blue | male=blue) = 78/114
P(female=blue | male=blue) = 0.68
c) What is the probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes?
As per the rule of conditional probability,
P(female=blue | male=brown) = 19/54
P(female=blue | male=brown) = 0.35
d) What is the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes?
As per the rule of conditional probability,
P(female=blue | male=green) = 11/36
P(female=blue | male=green) = 0.31
e) Does it appear that the eye colors of male respondents and their partners are independent? Explain
If the following relation holds true then we can conclude that the eye colors of male respondents and their partners are independent.
∵ P(B | A) = P(B)
P(female=blue | male=brown) = P(female=blue)
or alternatively, you can also test
P(female=blue | male=green) = P(female=blue)
P(female=blue | male=blue) = P(female=blue)
But
P(female=blue | male=brown) ≠ P(female=blue)
19/54 ≠ 108/204
0.35 ≠ 0.53
Therefore, we can conclude that the eye colors of male respondents and their partners are not independent.
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