IDNLearn.com offers a comprehensive platform for finding and sharing knowledge. Our experts are available to provide in-depth and trustworthy answers to any questions you may have.
Sagot :
To find the probability that a randomly selected student from the class takes Consumer Education, French, or both, we need to use the principle of inclusion-exclusion. Here's the step-by-step solution:
1. Determine the total number of students:
Total students = 80
2. Calculate individual probabilities:
- Number of students taking Consumer Education = 22
[tex]\( P(\text{Consumer Education}) = \frac{22}{80} \)[/tex]
- Number of students taking French = 20
[tex]\( P(\text{French}) = \frac{20}{80} \)[/tex]
- Number of students taking both Consumer Education and French = 4
[tex]\( P(\text{Both}) = \frac{4}{80} \)[/tex]
3. Convert these probabilities to decimals (this step is only for clarity, we can use fractions directly in calculations):
[tex]\( P(\text{Consumer Education}) = \frac{22}{80} = 0.275 \)[/tex]
[tex]\( P(\text{French}) = \frac{20}{80} = 0.25 \)[/tex]
[tex]\( P(\text{Both}) = \frac{4}{80} = 0.05 \)[/tex]
4. Apply the inclusion-exclusion principle:
The formula to find the probability of either event (taking Consumer Education, taking French, or taking both) is:
[tex]\[ P(\text{Consumer Education or French}) = P(\text{Consumer Education}) + P(\text{French}) - P(\text{Both}) \][/tex]
Plugging in the numbers:
[tex]\[ P(\text{Consumer Education or French}) = 0.275 + 0.25 - 0.05 = 0.475 \][/tex]
5. Convert the result to a fraction (for comparison with given options):
To match the given options, let's express each probability with a common denominator. First, note that the common denominator for the fractions provided is 40.
- [tex]\( P(\text{Consumer Education}) = \frac{22}{80} = \frac{11}{40} \)[/tex]
- [tex]\( P(\text{French}) = \frac{20}{80} = \frac{1}{4} = \frac{10}{40} \)[/tex]
- [tex]\( P(\text{Both}) = \frac{4}{80} = \frac{1}{20} = \frac{2}{40} \)[/tex]
Now, use these fractions in the inclusion-exclusion formula:
[tex]\[ P = \frac{11}{40} + \frac{10}{40} - \frac{2}{40} = \frac{11 + 10 - 2}{40} = \frac{19}{40} = 0.475 \][/tex]
6. Compare with given options:
- [tex]\( P = \frac{11}{40} + \frac{1}{4} - \frac{1}{20} = 0.475 \)[/tex] (matches our 0.475)
- [tex]\( P = \frac{11}{40} + \frac{1}{4} - \frac{1}{10} \approx 0.425 \)[/tex]
- [tex]\( P = \frac{11}{40} + \frac{1}{4} \approx 0.525 \)[/tex]
- [tex]\( P = \frac{11}{40} + \frac{1}{4} + \frac{1}{20} \approx 0.575 \)[/tex]
Therefore, the correct equation that can be used to find the probability [tex]\( P \)[/tex] is:
[tex]\[ P = \frac{11}{40} + \frac{1}{4} - \frac{1}{20} \][/tex]
1. Determine the total number of students:
Total students = 80
2. Calculate individual probabilities:
- Number of students taking Consumer Education = 22
[tex]\( P(\text{Consumer Education}) = \frac{22}{80} \)[/tex]
- Number of students taking French = 20
[tex]\( P(\text{French}) = \frac{20}{80} \)[/tex]
- Number of students taking both Consumer Education and French = 4
[tex]\( P(\text{Both}) = \frac{4}{80} \)[/tex]
3. Convert these probabilities to decimals (this step is only for clarity, we can use fractions directly in calculations):
[tex]\( P(\text{Consumer Education}) = \frac{22}{80} = 0.275 \)[/tex]
[tex]\( P(\text{French}) = \frac{20}{80} = 0.25 \)[/tex]
[tex]\( P(\text{Both}) = \frac{4}{80} = 0.05 \)[/tex]
4. Apply the inclusion-exclusion principle:
The formula to find the probability of either event (taking Consumer Education, taking French, or taking both) is:
[tex]\[ P(\text{Consumer Education or French}) = P(\text{Consumer Education}) + P(\text{French}) - P(\text{Both}) \][/tex]
Plugging in the numbers:
[tex]\[ P(\text{Consumer Education or French}) = 0.275 + 0.25 - 0.05 = 0.475 \][/tex]
5. Convert the result to a fraction (for comparison with given options):
To match the given options, let's express each probability with a common denominator. First, note that the common denominator for the fractions provided is 40.
- [tex]\( P(\text{Consumer Education}) = \frac{22}{80} = \frac{11}{40} \)[/tex]
- [tex]\( P(\text{French}) = \frac{20}{80} = \frac{1}{4} = \frac{10}{40} \)[/tex]
- [tex]\( P(\text{Both}) = \frac{4}{80} = \frac{1}{20} = \frac{2}{40} \)[/tex]
Now, use these fractions in the inclusion-exclusion formula:
[tex]\[ P = \frac{11}{40} + \frac{10}{40} - \frac{2}{40} = \frac{11 + 10 - 2}{40} = \frac{19}{40} = 0.475 \][/tex]
6. Compare with given options:
- [tex]\( P = \frac{11}{40} + \frac{1}{4} - \frac{1}{20} = 0.475 \)[/tex] (matches our 0.475)
- [tex]\( P = \frac{11}{40} + \frac{1}{4} - \frac{1}{10} \approx 0.425 \)[/tex]
- [tex]\( P = \frac{11}{40} + \frac{1}{4} \approx 0.525 \)[/tex]
- [tex]\( P = \frac{11}{40} + \frac{1}{4} + \frac{1}{20} \approx 0.575 \)[/tex]
Therefore, the correct equation that can be used to find the probability [tex]\( P \)[/tex] is:
[tex]\[ P = \frac{11}{40} + \frac{1}{4} - \frac{1}{20} \][/tex]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.
