Answer :
Let's use the given data from the table.
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \# \text{ of siblings} & 0-2 & 3-5 & 6 \text{ or more} & \text{Total} \\ \hline \text{Broncos Fan} & 6 & 18 & 11 & 35 \\ \hline \text{Not a Broncos Fan} & 1 & 7 & 4 & 12 \\ \hline \text{Total} & 7 & 25 & 15 & 47 \\ \hline \end{array} \][/tex]
### a) Probability that the person was not a Broncos fan
The total number of people is [tex]\(47\)[/tex], and the number of people who are not Broncos fans is [tex]\(12\)[/tex]. The probability is given by:
[tex]\[ P(\text{Not a Broncos fan}) = \frac{\text{Number of Not Broncos Fans}}{\text{Total Number of People}} = \frac{12}{47} \approx 0.2553 \][/tex]
### b) Given that a person is a Broncos fan, what is the probability that they have 6 or more siblings?
The total number of Broncos fans is [tex]\(35\)[/tex], and the number of Broncos fans with 6 or more siblings is [tex]\(11\)[/tex]. The conditional probability is:
[tex]\[ P(\text{6 or more siblings} \mid \text{Broncos Fan}) = \frac{\text{Number of Broncos Fans with 6 or more siblings}}{\text{Total Number of Broncos Fans}} = \frac{11}{35} \approx 0.3143 \][/tex]
### c) Given that a person has 3-5 siblings, what is the probability that the person is not a Broncos fan?
The total number of people with 3-5 siblings is [tex]\(25\)[/tex], and the number of those who are not Broncos fans is [tex]\(7\)[/tex]. The conditional probability is:
[tex]\[ P(\text{Not a Broncos Fan} \mid \text{3-5 siblings}) = \frac{\text{Number of Not Broncos Fans with 3-5 siblings}}{\text{Total Number of People with 3-5 siblings}} = \frac{7}{25} = 0.28 \][/tex]
### d) Probability that the person was a Broncos fan
The total number of people is [tex]\(47\)[/tex], and the number of Broncos fans is [tex]\(35\)[/tex]. The probability is:
[tex]\[ P(\text{Broncos Fan}) = \frac{\text{Number of Broncos Fans}}{\text{Total Number of People}} = \frac{35}{47} \approx 0.7447 \][/tex]
### e) Probability that the person is not a Broncos fan and has 3-5 siblings
The number of people who are not Broncos fans and have 3-5 siblings is [tex]\(7\)[/tex], and the total number of people is [tex]\(47\)[/tex]. The probability is:
[tex]\[ P(\text{Not a Broncos Fan and 3-5 siblings}) = \frac{\text{Number of Not Broncos Fans with 3-5 siblings}}{\text{Total Number of People}} = \frac{7}{47} \approx 0.1489 \][/tex]
### f) Probability that the person is not a Broncos fan or has 0-2 siblings
To find the probability that a person is not a Broncos fan or has 0-2 siblings, we use the principle of inclusion-exclusion:
[tex]\[ P(\text{Not a Broncos Fan or 0-2 siblings}) = P(\text{Not a Broncos Fan}) + P(\text{0-2 siblings}) - P(\text{Not a Broncos Fan and 0-2 siblings}) \][/tex]
We have already calculated [tex]\(P(\text{Not a Broncos Fan}) = \frac{12}{47}\)[/tex]. The probability of having 0-2 siblings is:
[tex]\[ P(\text{0-2 siblings}) = \frac{7}{47} \][/tex]
The probability of being not a Broncos fan and having 0-2 siblings is:
[tex]\[ P(\text{Not a Broncos Fan and 0-2 siblings}) = \frac{1}{47} \][/tex]
Now, apply the formula:
[tex]\[ P(\text{Not a Broncos Fan or 0-2 siblings}) = \frac{12}{47} + \frac{7}{47} - \frac{1}{47} = \frac{18}{47} \approx 0.3830 \][/tex]
### g) Probability that the person was a Broncos fan and has 3-5 siblings
The number of Broncos fans with 3-5 siblings is [tex]\(18\)[/tex], and the total number of people is [tex]\(47\)[/tex]. The probability is:
[tex]\[ P(\text{Broncos Fan and 3-5 siblings}) = \frac{\text{Number of Broncos Fans with 3-5 siblings}}{\text{Total Number of People}} = \frac{18}{47} \approx 0.3830 \][/tex]
Here are the final answers in a pristine form:
- a) [tex]\(0.2553\)[/tex]
- b) [tex]\(0.3143\)[/tex]
- c) [tex]\(0.28\)[/tex]
- d) [tex]\(0.7447\)[/tex]
- e) [tex]\(0.1489\)[/tex]
- f) [tex]\(0.3830\)[/tex]
- g) [tex]\(0.3830\)[/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline \# \text{ of siblings} & 0-2 & 3-5 & 6 \text{ or more} & \text{Total} \\ \hline \text{Broncos Fan} & 6 & 18 & 11 & 35 \\ \hline \text{Not a Broncos Fan} & 1 & 7 & 4 & 12 \\ \hline \text{Total} & 7 & 25 & 15 & 47 \\ \hline \end{array} \][/tex]
### a) Probability that the person was not a Broncos fan
The total number of people is [tex]\(47\)[/tex], and the number of people who are not Broncos fans is [tex]\(12\)[/tex]. The probability is given by:
[tex]\[ P(\text{Not a Broncos fan}) = \frac{\text{Number of Not Broncos Fans}}{\text{Total Number of People}} = \frac{12}{47} \approx 0.2553 \][/tex]
### b) Given that a person is a Broncos fan, what is the probability that they have 6 or more siblings?
The total number of Broncos fans is [tex]\(35\)[/tex], and the number of Broncos fans with 6 or more siblings is [tex]\(11\)[/tex]. The conditional probability is:
[tex]\[ P(\text{6 or more siblings} \mid \text{Broncos Fan}) = \frac{\text{Number of Broncos Fans with 6 or more siblings}}{\text{Total Number of Broncos Fans}} = \frac{11}{35} \approx 0.3143 \][/tex]
### c) Given that a person has 3-5 siblings, what is the probability that the person is not a Broncos fan?
The total number of people with 3-5 siblings is [tex]\(25\)[/tex], and the number of those who are not Broncos fans is [tex]\(7\)[/tex]. The conditional probability is:
[tex]\[ P(\text{Not a Broncos Fan} \mid \text{3-5 siblings}) = \frac{\text{Number of Not Broncos Fans with 3-5 siblings}}{\text{Total Number of People with 3-5 siblings}} = \frac{7}{25} = 0.28 \][/tex]
### d) Probability that the person was a Broncos fan
The total number of people is [tex]\(47\)[/tex], and the number of Broncos fans is [tex]\(35\)[/tex]. The probability is:
[tex]\[ P(\text{Broncos Fan}) = \frac{\text{Number of Broncos Fans}}{\text{Total Number of People}} = \frac{35}{47} \approx 0.7447 \][/tex]
### e) Probability that the person is not a Broncos fan and has 3-5 siblings
The number of people who are not Broncos fans and have 3-5 siblings is [tex]\(7\)[/tex], and the total number of people is [tex]\(47\)[/tex]. The probability is:
[tex]\[ P(\text{Not a Broncos Fan and 3-5 siblings}) = \frac{\text{Number of Not Broncos Fans with 3-5 siblings}}{\text{Total Number of People}} = \frac{7}{47} \approx 0.1489 \][/tex]
### f) Probability that the person is not a Broncos fan or has 0-2 siblings
To find the probability that a person is not a Broncos fan or has 0-2 siblings, we use the principle of inclusion-exclusion:
[tex]\[ P(\text{Not a Broncos Fan or 0-2 siblings}) = P(\text{Not a Broncos Fan}) + P(\text{0-2 siblings}) - P(\text{Not a Broncos Fan and 0-2 siblings}) \][/tex]
We have already calculated [tex]\(P(\text{Not a Broncos Fan}) = \frac{12}{47}\)[/tex]. The probability of having 0-2 siblings is:
[tex]\[ P(\text{0-2 siblings}) = \frac{7}{47} \][/tex]
The probability of being not a Broncos fan and having 0-2 siblings is:
[tex]\[ P(\text{Not a Broncos Fan and 0-2 siblings}) = \frac{1}{47} \][/tex]
Now, apply the formula:
[tex]\[ P(\text{Not a Broncos Fan or 0-2 siblings}) = \frac{12}{47} + \frac{7}{47} - \frac{1}{47} = \frac{18}{47} \approx 0.3830 \][/tex]
### g) Probability that the person was a Broncos fan and has 3-5 siblings
The number of Broncos fans with 3-5 siblings is [tex]\(18\)[/tex], and the total number of people is [tex]\(47\)[/tex]. The probability is:
[tex]\[ P(\text{Broncos Fan and 3-5 siblings}) = \frac{\text{Number of Broncos Fans with 3-5 siblings}}{\text{Total Number of People}} = \frac{18}{47} \approx 0.3830 \][/tex]
Here are the final answers in a pristine form:
- a) [tex]\(0.2553\)[/tex]
- b) [tex]\(0.3143\)[/tex]
- c) [tex]\(0.28\)[/tex]
- d) [tex]\(0.7447\)[/tex]
- e) [tex]\(0.1489\)[/tex]
- f) [tex]\(0.3830\)[/tex]
- g) [tex]\(0.3830\)[/tex]
