Question

In a class of 325 juniors, 190 take Algebra 2 and 165 take Chemistry. There are 40 students who take neither. What is the probability that a student chosen randomly from the junior class takes Algebra 2 AND Chemistry?

Question 3 options:

14/65


8/73


71/65


63/65

Answer 1

To find the probability that a randomly chosen student takes both Algebra 2 and Chemistry, we can use the principle of inclusion-exclusion to find the number of students who take at least one of the subjects.

1. **Total number of students**: 325
2. **Students taking Algebra 2**: 190
3. **Students taking Chemistry**: 165
4. **Students taking neither**: 40

First, we find the number of students who take at least one of the subjects:

\[
\text{Students taking at least one subject} = \text{Total students} - \text{Students taking neither}
\]

\[
= 325 - 40 = 285
\]

Now, using the principle of inclusion-exclusion, we can express the number of students taking at least one subject:

\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]

where:
- \( n(A) \) is the number of students taking Algebra 2 (190)
- \( n(B) \) is the number of students taking Chemistry (165)
- \( n(A \cup B) \) is the number of students taking at least one of the subjects (285)
- \( n(A \cap B) \) is the number of students taking both subjects (which we want to find)

Substituting the known values into the inclusion-exclusion formula gives us:

\[
285 = 190 + 165 - n(A \cap B)
\]

Simplifying this:

\[
285 = 355 - n(A \cap B)
\]

Rearranging gives:

\[
n(A \cap B) = 355 - 285 = 70
\]

So, 70 students take both Algebra 2 and Chemistry.

Finally, we want to find the probability that a randomly chosen student takes both subjects:

\[
P(A \cap B) = \frac{n(A \cap B)}{\text{Total students}} = \frac{70}{325}
\]

Now, we simplify \( \frac{70}{325} \):

Both 70 and 325 can be divided by 5:

\[
\frac{70 \div 5}{325 \div 5} = \frac{14}{65}
\]

Thus, the probability that a student chosen randomly from the junior class takes both Algebra 2 and Chemistry is:

\[
\boxed{\frac{14}{65}}
\]