Grades on a very large statistics course have historically been awarded according to the following distribution.

HD D C P Z or Fail
0.15 0.20 0.30 0.30 0.05

What is the probability that two students picked independent of each other and at random both get an HD?

Question

Grades on a very large statistics course have historically been awarded according to the following distribution.

HD	D	C	P	Z or Fail
0.15	0.20	0.30	0.30	0.05

What is the probability that two students picked independent of each other and at random both get an HD?

BlackTom AI · Accepted Answer

For independent events, multiply their probabilities: P(both get HD) = P(first is HD) × P(second is HD).

1. Relevant knowledge points and concepts
- Probability of an event expressed as a decimal or fraction (15% = 0.15).
- Independence: when two selections are independent, the joint probability equals the product of individual probabilities.
- Multiplication rule for independent events: P(A and B) = P(A)·P(B).
- The phrase "very large statistics course" implies sampling without noticeable change in probabilities, so independence is valid.

2. Complete reasoning process step by step
- Given P(student gets HD) = 0.15.
- Two students picked independently: the event that both get HD is the intersection of two independent events.
- Use the multiplication rule: P(both HD) = 0.15 × 0.15.

3. Calculations with intermediate steps
- Convert percent to fraction: 15% = 15/100 = 3/20.
- Square the probability: (3/20)² = 9/400.
- Convert back to decimal: 9 ÷ 400 = 0.0225.
- Or directly: 0.15 × 0.15 = 0.0225.

4. Why the correct answer is correct
- Multiplying the independent probabilities gives the exact probability that both independent events occur. The arithmetic (0.15²) yields 0.0225, so there is a 2.25% chance both randomly chosen students receive an HD.

5. Why other options would be incorrect
- Any answer that adds probabilities (e.g., 0.15 + 0.15 = 0.30) would be incorrect because addition is for mutually exclusive events, not joint independent events.
- Any other product (e.g., using 0.30 or 0.05) would use the wrong single-event probability.

6. Conclusion
The probability that two independently and randomly picked students both get an HD is 0.0225, which matches the provided correct answer.

Grades on a very large statistics course have historically been awarded according to the following distribution. HD D C P Z or Fail 0.15 0.20 0.30 0.30 0.05 What is the probability that two students picked independent of each other and at random both get an HD?单项选择题

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