To determine the probability, [tex]\(P\)[/tex], that a randomly selected student from this class takes either Consumer Education, French, or both, we can use the principle of inclusion and exclusion. We will follow the steps below for a detailed, step-by-step explanation:
1. Define the Total Number of Students:
The total number of students in the class is given as [tex]\(80\)[/tex].
2. Students Taking Consumer Education:
The number of students taking Consumer Education is [tex]\(22\)[/tex]. The probability that a randomly selected student takes Consumer Education is:
[tex]\[
\frac{22}{80}
\][/tex]
3. Students Taking French:
The number of students taking French is [tex]\(20\)[/tex]. The probability that a randomly selected student takes French is:
[tex]\[
\frac{20}{80}
\][/tex]
4. Students Taking Both Subjects:
The number of students taking both Consumer Education and French is [tex]\(4\)[/tex]. The probability that a randomly selected student takes both subjects is:
[tex]\[
\frac{4}{80}
\][/tex]
5. Apply the Principle of Inclusion and Exclusion:
To find the probability that a student takes at least one of the subjects (Consumer Education, French, or both), we need to add the probabilities of the individual subjects and then subtract the probability of the overlap (students taking both subjects), to avoid double-counting those students:
[tex]\[
P = \left( \frac{22}{80} \right) + \left( \frac{20}{80} \right) - \left( \frac{4}{80} \right)
\][/tex]
6. Convert the Probabilities to Simplified Fractions:
Simplify each fraction involved:
[tex]\[
\frac{22}{80} = \frac{11}{40}, \quad \frac{20}{80} = \frac{1}{4}, \quad \text{and} \quad \frac{4}{80} = \frac{1}{20}
\][/tex]
7. Combine the Probabilities:
Substituting these simplified fractions into the inclusion-exclusion formula gives:
[tex]\[
P = \frac{11}{40} + \frac{1}{4} - \frac{1}{20}
\][/tex]
Thus, the equation that can be used to find the probability, [tex]\(P\)[/tex], that a randomly selected student from this class takes Consumer Education, French, or both is:
[tex]\[
P = \frac{11}{40} + \frac{1}{4} - \frac{1}{20}
\][/tex]
