Answer :
Certainly! Let's break down the problem step by step according to each part.
### Part (a) Probability that the person is a Republican given that the person is Male
The given information tells us:
- The number of male Republicans is 61.
- The total number of males is 138.
To find the probability that a person is a Republican given that the person is Male, we use the formula for conditional probability:
[tex]\[ P(R|M) = \frac{P(R \cap M)}{P(M)} \][/tex]
Where:
- [tex]\( P(R \cap M) \)[/tex] is the probability that the person is both Republican and Male. This is the ratio of male Republicans to the total population, which is 61.
- [tex]\( P(M) \)[/tex] is the probability that the person is Male. This is the total number of males divided by the total population, which is 138.
Thus, the probability is:
[tex]\[ P(R|M) = \frac{61}{138} \][/tex]
Simplifying the fraction is not necessary, as the problem provides a specific probability. The numerical result is:
[tex]\[ P(R|M) \approx 0.442 \][/tex]
### Part (b) Probability that the person is Male given that the person is a Democrat
The information provided includes:
- The total number of males is 138.
- The total number of Democrats is 89.
To find the probability that a person is Male given that the person is a Democrat, we use a similar conditional probability formula:
[tex]\[ P(M|D) = \frac{P(M \cap D)}{P(D)} \][/tex]
Where:
- [tex]\( P(M \cap D) \)[/tex] is the probability of being both Male and a Democrat, which is the total number of males (138) because it's comparing males across the entire population.
- [tex]\( P(D) \)[/tex] is the probability of being a Democrat, which is 89.
So, the probability is:
[tex]\[ P(M|D) = \frac{138}{213} \][/tex]
Again, this fraction shows the specific relationship. The numerical result is:
[tex]\[ P(M|D) \approx 0.648 \][/tex]
### Part (c) Probability that the person is Female given that the person is an Independent
For this part, we need to know:
- The number of female Independents is 17.
- The total number of Independents is 27.
Using the conditional probability formula:
[tex]\[ P(F|I) = \frac{P(F \cap I)}{P(I)} \][/tex]
Where:
- [tex]\( P(F \cap I) \)[/tex] is the probability of being both Female and Independent, which is the total number of female Independents (17).
- [tex]\( P(I) \)[/tex] is the probability of being an Independent, which is 27.
Therefore, the probability is:
[tex]\[ P(F|I) = \frac{17}{27} \][/tex]
And the numerical result is:
[tex]\[ P(F|I) \approx 0.630 \][/tex]
To summarize:
(a) The probability that the person is a Republican given that the person is Male is approximately 0.442.
(b) The probability that the person is Male given that the person is a Democrat is approximately 0.648.
(c) The probability that the person is Female given that the person is an Independent is approximately 0.630.
### Part (a) Probability that the person is a Republican given that the person is Male
The given information tells us:
- The number of male Republicans is 61.
- The total number of males is 138.
To find the probability that a person is a Republican given that the person is Male, we use the formula for conditional probability:
[tex]\[ P(R|M) = \frac{P(R \cap M)}{P(M)} \][/tex]
Where:
- [tex]\( P(R \cap M) \)[/tex] is the probability that the person is both Republican and Male. This is the ratio of male Republicans to the total population, which is 61.
- [tex]\( P(M) \)[/tex] is the probability that the person is Male. This is the total number of males divided by the total population, which is 138.
Thus, the probability is:
[tex]\[ P(R|M) = \frac{61}{138} \][/tex]
Simplifying the fraction is not necessary, as the problem provides a specific probability. The numerical result is:
[tex]\[ P(R|M) \approx 0.442 \][/tex]
### Part (b) Probability that the person is Male given that the person is a Democrat
The information provided includes:
- The total number of males is 138.
- The total number of Democrats is 89.
To find the probability that a person is Male given that the person is a Democrat, we use a similar conditional probability formula:
[tex]\[ P(M|D) = \frac{P(M \cap D)}{P(D)} \][/tex]
Where:
- [tex]\( P(M \cap D) \)[/tex] is the probability of being both Male and a Democrat, which is the total number of males (138) because it's comparing males across the entire population.
- [tex]\( P(D) \)[/tex] is the probability of being a Democrat, which is 89.
So, the probability is:
[tex]\[ P(M|D) = \frac{138}{213} \][/tex]
Again, this fraction shows the specific relationship. The numerical result is:
[tex]\[ P(M|D) \approx 0.648 \][/tex]
### Part (c) Probability that the person is Female given that the person is an Independent
For this part, we need to know:
- The number of female Independents is 17.
- The total number of Independents is 27.
Using the conditional probability formula:
[tex]\[ P(F|I) = \frac{P(F \cap I)}{P(I)} \][/tex]
Where:
- [tex]\( P(F \cap I) \)[/tex] is the probability of being both Female and Independent, which is the total number of female Independents (17).
- [tex]\( P(I) \)[/tex] is the probability of being an Independent, which is 27.
Therefore, the probability is:
[tex]\[ P(F|I) = \frac{17}{27} \][/tex]
And the numerical result is:
[tex]\[ P(F|I) \approx 0.630 \][/tex]
To summarize:
(a) The probability that the person is a Republican given that the person is Male is approximately 0.442.
(b) The probability that the person is Male given that the person is a Democrat is approximately 0.648.
(c) The probability that the person is Female given that the person is an Independent is approximately 0.630.
