To solve the problem of finding the probability that a student picked at random from the school choir is either a girl or in seventh grade, we need to follow several steps systematically.
1. Identify the total number of students: We are given that the total number of students in the choir is 80.
2. Identify the number of students in the seventh grade: We are given that there are 36 students in the seventh grade.
3. Identify the number of girls: We know that there are 40 girls in the choir.
4. Identify the number of girls in the seventh grade: We are given that there are 18 girls in the seventh grade.
5. Calculate the number of students who are either a girl or in the seventh grade:
- We will use the inclusion-exclusion principle which states that the number of elements in the union of two sets is the sum of the elements in each set minus the number of elements in their intersection.
- Let [tex]\(G\)[/tex] be the set of all girls, and [tex]\(S\)[/tex] be the set of all seventh grade students. Then, the number of students who are either girls or in the seventh grade is given by:
[tex]\[
|G \cup S| = |G| + |S| - |G \cap S|
\][/tex]
- Plugging in the numbers from the problem:
[tex]\[
|G \cup S| = 40 + 36 - 18
\][/tex]
[tex]\[
|G \cup S| = 58
\][/tex]
6. Calculate the probability that a randomly chosen student is either a girl or in the seventh grade:
- The probability is given by the number of favorable outcomes divided by the total number of possible outcomes.
- Here, the number of favorable outcomes is the number of students who are either girls or in the seventh grade, which is 58, and the total number of possible outcomes is the total number of students, which is 80.
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
\][/tex]
[tex]\[
\text{Probability} = \frac{58}{80}
\][/tex]
7. Simplify the fraction:
- To simplify [tex]\(\frac{58}{80}\)[/tex], we can divide both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{58 \div 2}{80 \div 2} = \frac{29}{40}
\][/tex]
Therefore, the probability that a student picked at random from the choir is either a girl or in seventh grade is [tex]\(\frac{29}{40}\)[/tex].
The correct answer is:
[tex]\[
\boxed{\frac{29}{40}}
\][/tex]
