Question

In a survey of 232 total people, the following data were obtained relating gender to color-blindness: Color-Blind (C) Not Color - Blind C Total Male (M) 96 67 163 Female (F) 15 54 69 Total 111 121 232A person is randomly selected. What is the probability that the person is: a. Male? b. Male and Color-blind? c. Male given that the person is Color-blind? d. Color-blind given that the person is Male? e. Female given that the person is not Color-blind? f. Are the events Male and Color blind independent?

Answer 1

The probabilities of a person being male, male and color-blind, and the conditional probabilities of male given color-blind, color-blind given male, and female given not color-blind were calculated from survey data. It was found that the events Male and Color blind are not independent. Color blindness is clarified as a sex-linked attribute, specifically X-linked recessive.

Step-by-step explanation:

Given survey data about gender and color-blindness in a population of 232 people, we can determine several probabilities.

• a. Probability of being male: This is the number of males (163) divided by the total survey population (232), which equals 0.703.
• b. Probability of being male and color-blind: This is the number of color-blind males (96) divided by the total population (232), which is roughly 0.414.
• c. Probability of being male given that the person is color-blind: This is the number of color-blind males (96) divided by the total number of color-blind people (111), which equals roughly 0.865.
• d. Probability of being color-blind given that the person is male: This is the number of color-blind males (96) divided by the total number of males (163), which is roughly 0.589.
• e. Probability of being female given that the person is not color-blind: This is the number of not color-blind females (54) divided by the total number of people who are not color-blind (121), which is approximately 0.446.

f. Are the events Male and Color blind independent? Two events, A and B, are independent if P(A)*P(B) = P(A and B). For being male (P(M) = 163/232) and being color-blind (P(C) = 111/232), if they are independent, P(M)*P(C) should equal P(M and C), which is 96/232. Calculating P(M)*P(C), we get approximately 0.703 * 0.478, which is not equal to roughly 0.414 (P(M and C)), implying that the events are not independent.

There is also a common misconception regarding color blindness. The Punnett square indicates that color blindness is a sex-linked trait, specifically X-linked recessive, and not autosomal dominant or recessive. Thus, males cannot be carriers; they are either color-blind or not since they have only one X chromosome.

Answer 2

The survey data requires calculating various probabilities relating to being male and color-blind. These probabilities are derived from the provided data, showing the likelihood of these traits in the sampled population. The data also illustrates the sex-linked nature of color-blindness.

Step-by-step explanation:

The question involves calculating probabilities from given survey data relating gender to color-blindness. Here are the solutions for each part:

1. Probability of being Male (P(M)): Total males / Total people = 163 / 232.
2. Probability of being Male and Color-blind (P(M ∩ C)): Males that are color-blind / Total people = 96 / 232.
3. Probability of being Male given that the person is Color-blind (P(M | C)): Males that are color-blind / Total color-blind = 96 / 111.
4. Probability of being Color-blind given that the person is Male (P(C | M)): Males that are color-blind / Total males = 96 / 163.
5. Probability of being Female given that the person is not Color-blind (P(F | not C)): Females that are not color-blind / Total not color-blind = 54 / 121.
6. To check for independence between being Male and Color-blind, compare P(M ∩ C) with P(M) × P(C). If these are equal, the events are independent; otherwise, they are not. In this case, they are not equal, so being Male and Color-blind are not independent events.

Color-blindness is a sex-linked trait, which explains the distribution in the given data. Males have only one X chromosome, so if they inherit the color-blind gene on the X chromosome, they will show the trait.

Females have two X chromosomes and are less likely to exhibit color-blindness as it is recessive and usually masked by the other normal X chromosome.