Question

How are your grades? In a recent semester at a local university, 700 students enrolled in both Statistics and Psychology . Of these students, 76 got an A in statistics, 72 got an A in psychology, and 45 got an A in both statistics and psychology. Round the answers to four decimal places, as needed.

(a) Find the probability that a randomly chosen student got an A in statistics or psychology or both.
(b) Find the probability that a randomly chosen student did not get an A in psychology.

Answer 1

To solve the problem, we will use the principles of probability and the concept of the union of events.

Let:
- \( A \): the event that a student got an A in Statistics
- \( B \): the event that a student got an A in Psychology

From the information provided:
- Total students (\( n \)) = 700
- \( n(A) \) = Number of students who got an A in Statistics = 76
- \( n(B) \) = Number of students who got an A in Psychology = 72
- \( n(A \cap B) \) = Number of students who got an A in both Statistics and Psychology = 45

### (a) Probability that a randomly chosen student got an A in Statistics or Psychology or both:

We use the formula for the union of two sets:

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

First, we need to find the individual probabilities:

\[
P(A) = \frac{n(A)}{n} = \frac{76}{700}
\]
\[
P(B) = \frac{n(B)}{n} = \frac{72}{700}
\]
\[
P(A \cap B) = \frac{n(A \cap B)}{n} = \frac{45}{700}
\]

Now we can substitute into the formula:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{76}{700} + \frac{72}{700} - \frac{45}{700}
\]

Calculating the probabilities:

\[
P(A \cup B) = \frac{76 + 72 - 45}{700} = \frac{103}{700}
\]

Finally, to get the probability:

\[
P(A \cup B) = \frac{103}{700} \approx 0.1471
\]

### (b) Probability that a randomly chosen student did not get an A in Psychology:

The event that a student did not get an A in Psychology is the complement of the event \( B \):

\[
P(B') = 1 - P(B)
\]

We already calculated \( P(B) \):

\[
P(B) = \frac{72}{700}
\]

Thus:

\[
P(B') = 1 - \frac{72}{700} = \frac{700 - 72}{700} = \frac{628}{700}
\]

Calculating this probability gives:

\[
P(B') = \frac{628}{700} \approx 0.8971
\]

### Summary of Answers:
(a) The probability that a randomly chosen student got an A in Statistics or Psychology or both is approximately **0.1471**.

(b) The probability that a randomly chosen student did not get an A in Psychology is approximately **0.8971**.