Let's analyze the given data for the survey involving 209 people. We have information about the gender and color-blindness of the respondents. The data can be summarized in the following table:
| | Color-Blind | Not Color-Blind | Total |
|----------|-------------|-----------------|-------|
| Male | 76 | 70 | 146 |
| Female | 10 | 53 | 63 |
| Total | 86 | 123 | 209 |
To answer the probability questions, we need to derive the requested probabilities step-by-step from the given data.
a) Probability that a person is male given that the person is color-blind:
This is a conditional probability. It can be calculated as follows:
[tex]\[
P(\text{Male}|\text{Color-blind}) = \frac{\text{Number of males who are color-blind}}{\text{Total number of color-blind people}}
\][/tex]
The number of males who are color-blind is 76 and the total number of color-blind people is 86. Thus,
[tex]\[
P(\text{Male}|\text{Color-blind}) = \frac{76}{86} = \frac{38}{43} \approx 0.883720930232558
\][/tex]
So, the given answer [tex]\(\frac{38}{43}\)[/tex] is correct.
b) Probability that a person is male and color-blind:
This is a joint probability, which can be calculated as:
[tex]\[
P(\text{Male and Color-blind}) = \frac{\text{Number of males who are color-blind}}{\text{Total number of people}}
\][/tex]
The number of males who are color-blind is 76 and the total number of people is 209. Thus,
[tex]\[
P(\text{Male and Color-blind}) = \frac{76}{209} \approx 0.36363636363636365
\][/tex]
So, the entire probability for event b) should be approximately [tex]\(0.36363636363636365\)[/tex].
c) Probability that a person is female given that the person is not color-blind:
This is another conditional probability. It can be calculated as follows:
[tex]\[
P(\text{Female}|\text{Not color-blind}) = \frac{\text{Number of females who are not color-blind}}{\text{Total number of not color-blind people}}
\][/tex]
The number of females who are not color-blind is 53 and the total number of not color-blind people is 123. Thus,
[tex]\[
P(\text{Female}|\text{Not color-blind}) = \frac{53}{123} \approx 0.43089430894308944
\][/tex]
The given answer [tex]\(\frac{21}{41}\approx 0.51219512195122\)[/tex] is incorrect, as the correct probability is around [tex]\(0.43089430894308944\)[/tex], not [tex]\( \frac{21}{41} \)[/tex].
Thus, for the problem:
- a) [tex]\(\frac{38}{43}\)[/tex] is correct.
- b) The probability should be around [tex]\(0.36363636363636365\)[/tex].
- c) [tex]\(\frac{21}{41}\)[/tex] is incorrect and the correct probability is [tex]\(\frac{53}{123} \approx 0.43089430894308944\)[/tex].
