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.

1 answer

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.