Hello, world!

Probability that a randomly chosen male respondent or his partner has blue eyes: This is the probability of the event B occurring in either the male or the female partner. We can calculate this by summing the probabilities of having blue eyes for males and females separately and then subtracting the double counting of those who are both male and female: P(B) = P(B_male) + P(B_female) - P(B_male and B_female) = (108/204) + (114/204) - (78/204) = (108 + 114 - 78) / 204 = 144 / 204 = 0.7059 So, the probability that a randomly chosen male respondent or his partner has blue eyes is approximately 0.7059. (b) Probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes: This is the conditional probability of having a partner with blue eyes given that the male respondent has blue eyes. We can calculate this using the formula for conditional probability: P(B_female | B_male) = P(B_male and B_female) / P(B_male) = (78/204) / (108/204) = 78 / 108 = 0.7222 So, the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes is approximately 0.7222. (c) Probability that a randomly chosen male respondent with brown/green eyes has a partner with blue eyes: We can similarly calculate these conditional probabilities using the same formula as in part (b). P(B_female | Br_male) = (19/204) / (55/204) = 19/55 ≈ 0.3455 P(B_female | G_male) = (11/204) / (41/204) = 11/41 ≈ 0.2683 So, the probability of a randomly chosen male respondent with brown eyes having a partner with blue eyes is approximately 0.3455, and for a male respondent with green eyes, it's approximately 0.2683. (d) To determine if the eye colors of male respondents and their partners are independent, we need to compare the joint probabilities with the product of marginal probabilities. If they are independent, the joint probabilities should equal the product of the marginal probabilities. For example, let's check for blue eyes: P(B_male and B_female) = 78 / 204 P(B_male) * P(B_female) = (108 / 204) * (114 / 204) If these two are approximately equal, we can say that male respondents' and their partners' eye colors are independent. Otherwise, they are dependent. We can perform this check for all combinations of eye colors. Let's carry out these calculations to confirm. Let's start by calculating the joint probabilities and product of marginal probabilities for blue eyes: Joint probability: P(B_male and B_female) = 78 / 204 Product of marginal probabilities: P(B_male) * P(B_female) = (108 / 204) * (114 / 204) Let's calculate: Joint probability: P(B_male and B_female) = 78 / 204 = 0.3824 Product of marginal probabilities: P(B_male) * P(B_female) = (108 / 204) * (114 / 204) ≈ 0.3703 These two values are close but not exactly equal, suggesting a slight dependence between male respondents' and their partners' eye colors.