For part (c), let’s analyze why the answer might be incorrect.

We know:

• The student has a 25% chance of passing each exam (probability of passing = 0.25 and failing = 0.75).
• She is allowed a maximum of three attempts.

The probability that she will take exactly three exams happens only if she fails the first two exams and then either passes or fails on the third attempt. We’ll calculate this probability step by step:

1. Probability of failing the first two attempts: This is $0.75 \times 0.75 = 0.5625$.
2. Probability of taking the third exam (exactly three attempts):
   • She must either pass on the third attempt (0.25 probability) or fail on the third attempt (0.75 probability).

Thus, the probability of taking the exam exactly three times is: $0.5625 \times 0.25 = 0.140625$

Since this answer of 0.140625 was initially marked incorrect, there might be a discrepancy in how the result was interpreted or calculated in the initial question's feedback. You might consider checking the problem setup or reviewing if rounding or formatting was specified as a potential factor in marking the answer.

Would you like additional clarification on tree diagrams, probability calculations, or further exploration of related probability questions?

---

Related Questions to Consider:

1. How would the tree diagram differ if the passing probability were higher, say 50%?
2. What would the probability of passing on the second attempt only be?
3. How does the maximum number of attempts allowed impact the probabilities?
4. If the student had unlimited attempts, how would the expected number of attempts change?
5. How would the results vary if each exam attempt had a different probability of passing?

Tip:

When calculating probabilities for multiple independent events, use tree diagrams to visually verify each path and outcome probability.