At a large university, the probability that a student takes calculus and is on the dean's list is .062 . The probability that a student is on the dean's list is 0.27 . Find the probability that the student is taking calculus, given that he or she is on the dean's list. Please round the final answer to 2 or  3 decimal places. Explain please P(the student is taking calculus given that he or she is on the deans list)


We are tasked with finding the conditional probability \( P(\text{Calculus} \mid \text{Dean's List}) \), which is the probability that a student is taking calculus given that they are on the Dean's List.

The formula for conditional probability is given by:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

Where:
- \( P(A \mid B) \) is the probability of event \( A \) occurring given that \( B \) has occurred.
- \( P(A \cap B) \) is the probability that both events \( A \) and \( B \) occur (i.e., the student takes calculus and is on the Dean's List).
- \( P(B) \) is the probability of event \( B \) occurring (i.e., the student is on the Dean's List).

Here:
- \( A \) is the event "the student is taking calculus."
- \( B \) is the event "the student is on the Dean's List."
- \( P(A \cap B) = 0.062 \) (the probability that a student takes calculus and is on the Dean's List).
- \( P(B) = 0.27 \) (the probability that a student is on the Dean's List).

Now, we can plug the values into the formula:

\[
P(\text{Calculus} \mid \text{Dean's List}) = \frac{0.062}{0.27}
\]

Let's calculate that.

\[
P(\text{Calculus} \mid \text{Dean's List}) = \frac{0.062}{0.27} \approx 0.2296
\]

Rounding to 2 or 3 decimal places, the probability is approximately:

\[
P(\text{Calculus} \mid \text{Dean's List}) \approx 0.230
\]

### Conclusion:
The probability that a student is taking calculus given that they are on the Dean's List is approximately 0.230 (or 23.0%).

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:
1. What is the interpretation of conditional probability in real-world scenarios?
2. How would you find \( P(\text{Dean's List} \mid \text{Calculus}) \)?
3. If \( P(\text{Dean's List}) \) was 0.3 instead of 0.27, how would the answer change?
4. What does the value of \( P(\text{Calculus} \mid \text{Dean's List}) \) tell us about the relationship between the two events?
5. How can we generalize this result to situations with more complex dependencies?

**Tip**: When solving conditional probability problems, always check whether events are independent, which simplifies calculations significantly.

At a large university, the probability that a student takes calculus and is on the dean's list is .062. The probability that a student is on the dean's list is 0.27. Find the probability that the student is taking calculus, given that he or she is on the dean's list. Please round the final answer to 2 or 3 decimal places. Explain please P(the student is taking calculus given that he or she is on the dean's list).

Learn how to find the probability that a student is taking calculus given they are on the dean's list. This example uses the conditional probability formula and involves real-world applications of probability. Ideal for students in Grades 10-12.

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