Practice a College Board-style free response question on the binomial distribution. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 2 · The Binomial Distribution
A basketball player makes 70% of her free throws, and her attempts can be treated as independent. In a practice session she takes 10 free throws. Let X be the number of free throws she makes.
A
Explain why X can be modeled with a binomial distribution, and state the values of n and p.
✓ Model answer
X meets all four conditions of a binomial setting (BINS): each free throw is Binary (made or missed), the attempts are Independent, there is a fixed Number of trials (n = 10), and the probability of success is the Same on each attempt (p = 0.7). So X follows a binomial distribution with n = 10 and p = 0.7.
Why it scores: Explicitly checks all four BINS conditions in context AND states n = 10 and p = 0.7. Listing "binomial" without verifying the conditions would not earn full credit.
B
Find the mean and standard deviation of X, and interpret the mean in context.
✓ Model answer
Mean = np = 10 × 0.7 = 7 free throws. Standard deviation = √(np(1 − p)) = √(10 × 0.7 × 0.3) = √2.1 ≈ 1.45 free throws. Interpretation: over many sessions of 10 attempts, she would make an average of about 7 free throws per session.
Why it scores: Uses np and √(np(1 − p)) correctly with units, AND interprets the mean as a long-run average in context. A bare number with no interpretation would lose the interpretation point.
C
Find the probability that she makes all 10 free throws, and the probability that she makes at least 9 of the 10. Show your work.
✓ Model answer
P(X = 10) = (0.7)^10 ≈ 0.028. For at least 9: P(X = 9) = C(10, 9)(0.7)^9(0.3)^1 = 10 × (0.7)^9 × 0.3 ≈ 0.121. So P(X ≥ 9) = P(X = 9) + P(X = 10) ≈ 0.121 + 0.028 = 0.149. She makes at least 9 of 10 about 15% of the time.
Why it scores: Computes P(X = 10) correctly, recognizes "at least 9" means X = 9 or X = 10, shows the binomial calculation for P(X = 9), and adds them. Forgetting that "at least 9" includes 10 is the most common error.
How to score points on AP Statistics FRQs
Check conditions in context. For a binomial, verify all four BINS conditions using the actual scenario, not just by name.
Interpret parameters, not just compute them. Explain what the mean means for this player and these attempts.
Translate "at least" and "at most" carefully. "At least 9" means X = 9 or X = 10 — include every case.
Show the probability setup. Write the combination and the powers of p and (1 − p); a bare number can lose credit.
Round sensibly and keep units. Report probabilities to about three decimals and label counts (free throws).