IDNLearn.com is your go-to resource for finding precise and accurate answers. Get step-by-step guidance for all your technical questions from our knowledgeable community members.
Sagot :
To solve this question, we break it down into two parts as specified:
### Part 1: Determine the Probability a Student is in Sports Given That They Are a Senior
This requires us to determine [tex]\( P(\text{Sports} \mid \text{Senior}) \)[/tex].
From the two-way table, observe the following:
- The total number of seniors is 35.
- The total number of seniors who are in sports is 25.
We apply the conditional probability formula:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{\text{Number of seniors in sports}}{\text{Total number of seniors}} \][/tex]
By using the numbers directly from the table:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{25}{35} \][/tex]
In simple mathematical terms:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{25}{35} = \frac{5}{7} \approx 0.714 \][/tex]
### Part 2: Determine the Probability That It's a Senior and in Sports
This requires us to determine [tex]\( P(\text{Senior and Sports}) \)[/tex].
From the provided data:
- The total number of students is 100.
- The number of seniors who are in sports is 25.
We apply the probability formula for combined events:
[tex]\[ P(\text{Senior and Sports}) = \frac{\text{Number of seniors in sports}}{\text{Total number of students}} \][/tex]
By using the numbers directly from the table:
[tex]\[ P(\text{Senior and Sports}) = \frac{25}{100} \][/tex]
Therefore:
[tex]\[ P(\text{Senior and Sports}) = \frac{25}{100} = \frac{1}{4} = 0.25 \][/tex]
### Summary of Results
1. The probability a student is in sports, given that they are a senior [tex]\( P(\text{Sports} \mid \text{Senior}) \)[/tex] is approximately 0.714.
2. The probability that it’s a senior in sports [tex]\( P(\text{Senior and Sports}) \)[/tex] is 0.25.
So, specifically for the given question:
[tex]\[ P(\text{Senior}) = 0.35 \][/tex]
[tex]\[ P(\text{Senior and Sports}) = 0.25 \][/tex]
These are the detailed solutions for the given probabilities.
### Part 1: Determine the Probability a Student is in Sports Given That They Are a Senior
This requires us to determine [tex]\( P(\text{Sports} \mid \text{Senior}) \)[/tex].
From the two-way table, observe the following:
- The total number of seniors is 35.
- The total number of seniors who are in sports is 25.
We apply the conditional probability formula:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{\text{Number of seniors in sports}}{\text{Total number of seniors}} \][/tex]
By using the numbers directly from the table:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{25}{35} \][/tex]
In simple mathematical terms:
[tex]\[ P(\text{Sports} \mid \text{Senior}) = \frac{25}{35} = \frac{5}{7} \approx 0.714 \][/tex]
### Part 2: Determine the Probability That It's a Senior and in Sports
This requires us to determine [tex]\( P(\text{Senior and Sports}) \)[/tex].
From the provided data:
- The total number of students is 100.
- The number of seniors who are in sports is 25.
We apply the probability formula for combined events:
[tex]\[ P(\text{Senior and Sports}) = \frac{\text{Number of seniors in sports}}{\text{Total number of students}} \][/tex]
By using the numbers directly from the table:
[tex]\[ P(\text{Senior and Sports}) = \frac{25}{100} \][/tex]
Therefore:
[tex]\[ P(\text{Senior and Sports}) = \frac{25}{100} = \frac{1}{4} = 0.25 \][/tex]
### Summary of Results
1. The probability a student is in sports, given that they are a senior [tex]\( P(\text{Sports} \mid \text{Senior}) \)[/tex] is approximately 0.714.
2. The probability that it’s a senior in sports [tex]\( P(\text{Senior and Sports}) \)[/tex] is 0.25.
So, specifically for the given question:
[tex]\[ P(\text{Senior}) = 0.35 \][/tex]
[tex]\[ P(\text{Senior and Sports}) = 0.25 \][/tex]
These are the detailed solutions for the given probabilities.
We are delighted to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.