AP Statistics (AP Stats): Binomial Distributions (BINS conditions, mean = np, SD = ?np(1?p))

AP Statistics – Binomial Distributions (BINS conditions, mean = np, SD = ?np(1?p))

AP Statistics: Binomial Distributions Study Guide

Exam-Ready in 20 Minutes

What This Is

A binomial distribution models the number of successes (X) in n independent trials, each with the same probability of success (p). It’s essential for the AP exam because it’s the foundation for proportion inference (confidence intervals and significance tests). Real-world example: A factory tests 50 light bulbs to estimate the proportion (p) that are defective. If 8 are defective, is this evidence that p > 0.10? Binomial distributions help answer this.

Key Terms & Formulas

• Binomial Setting (BINS):
• Binary outcomes (success/failure),
• Independent trials,
• Number of trials (n) is fixed,

• Same probability of success (p) for each trial. Example: Flipping a coin 10 times (n=10) to count heads (p=0.5).

• Binomial Random Variable (X): Count of successes in n trials. X ~ B(n, p).

• Mean (Expected Value) of X: ? = np

• n = number of trials, p = probability of success.

• Standard Deviation of X: ? = ?[np(1?p)]

• Measures spread of the distribution.

• Probability of Exactly k Successes: P(X=k) = binompdf(n, p, k) (TI-84)

• Use for "exactly" probabilities (e.g., P(X=3)).

• Probability of k or Fewer Successes: P(X?k) = binomcdf(n, p, k) (TI-84)

• Use for "at most" or "no more than" probabilities (e.g., P(X?5)).

• 10% Condition: If sampling without replacement, n-0.10N (where N = population size).

• Ensures trials are nearly independent.

• Large Counts Condition: np-10 and n(1?p)-10.

• Allows normal approximation to the binomial (use z-scores).

• Normal Approximation to Binomial: If Large Counts holds, X-N(np, ?[np(1?p)]).

• Convert X to z-score: z = (X-np) / ?[np(1?p)].

• Geometric vs. Binomial:

• Binomial: Fixed n trials, count successes.
• Geometric: Count trials until first success (no fixed n).

Step-by-Step / Process Flow

How to Solve a Binomial FRQ:
1. Define the Random Variable: - X = number of [successes] in [context]. X ~ B(n, p). - Example: X = number of defective bulbs in 50 trials. X ~ B(50, 0.10).

1. Check BINS Conditions:
2. Binary? (Success = defective, failure = not defective.)
3. Independent? (10% condition: 50-0.10*1000 = 100 bulbs in factory.)
4. N fixed? (50 bulbs tested.)

5. Same p? (Each bulb has 10% chance of being defective.)

6. Calculate Probabilities (if asked):

7. Use binompdf for "exactly" or binomcdf for "at most."

8. Example: P(X-8) = 1-P(X-7) = 1-binomcdf(50, 0.10, 7).

9. Find Mean and SD (if asked):

10. ? = np = 50*0.10 = 5.

11. = ?[np(1?p)] = ?[500.100.90]-2.12.

12. Normal Approximation (if Large Counts holds):

13. Check: np = 5-10? No-cannot use normal approximation.

14. If np-10 and n(1?p)-10, convert to z-score and use normalcdf.

15. Interpret in Context:

16. Example: "There is a 12.5% chance of finding 8 or more defective bulbs if the true defect rate is 10%."

Common Mistakes

• Mistake: Forgetting to check the 10% condition when sampling without replacement.

• Correction: Always verify n-0.10N for independence. Why? Without replacement, p changes slightly per trial.

• Mistake: Using binompdf for "at least" probabilities.

• Correction: For P(X-k), use 1-binomcdf(n, p, k?1)*. *Why?*binomcdf` gives P(X-k), so subtract from 1.

• Mistake: Applying normal approximation when np < 10 or n(1?p) < 10.

• Correction: Only use normal approximation if both np-10 and n(1?p)-10. Why? The binomial distribution is skewed if p is near 0 or 1.

• Mistake: Confusing binomial (fixed n) with geometric (trials until first success).

• Correction: Binomial counts successes in n trials; geometric counts trials until first success.

• Mistake: Misinterpreting p as the sample proportion (p?).

• Correction: p = true population proportion; p? = sample proportion (X/n).

AP Exam Insights

• Tricky Distinction: The exam loves to test normal approximation vs. exact binomial probabilities. Always check Large Counts!

• Example FRQ: "Can we use a normal approximation to calculate P(X-15) if n=100 and p=0.10?"-No (np=10, but n(1?p)=90-10; both must be-10).

• Calculator Pitfalls:

• binomcdf(n, p, k) gives P(X-k), not P(X < k). For P(X < k), use binomcdf(n, p, k?1).

• For P(X-k), use 1-`binomcdf(n, p, k?1)*.

• Common FRQ Setups:

• "Is there convincing evidence that p > [value]?"-Use binomial probability to find P(X-observed) and compare to ?.

• "What is the probability of [exactly/at least] k successes?"-Use binompdf/binomcdf.

• Context Matters: Always label X and define p in context. The AP graders deduct points for generic answers.

Quick Check Questions

1. MC: A basketball player makes 80% of her free throws. In a game, she takes 12 free throws. What is the probability she makes exactly 10?
2. (A) binompdf(12, 0.8, 10)
3. (B) binomcdf(12, 0.8, 10)
4. (C) binompdf(12, 0.2, 10)

5. (D) 1-binomcdf(12, 0.8, 9) Answer: (A) binompdf(12, 0.8, 10) gives P(X=10).

6. FRQ: A factory claims 5% of its widgets are defective. In a sample of 200 widgets, 18 are defective.

7. (a) Define the random variable and check BINS.
8. (b) Find the mean and SD of the number of defective widgets.
9. (c) Is it unusual to find 18 or more defective widgets if the claim is true? Justify with a probability. Answers:
10. (a) X = number of defective widgets in 200. BINS: Binary (defective/not), Independent (200-0.104000=400), n=200 fixed, p*=0.05 same.
11. (b)-= np = 2000.05 = 10;-= ?[2000.05*0.95]-3.08.

12. (c) P(X-18) = 1-P(X-17) = 1-binomcdf(200, 0.05, 17)-0.016. Since 0.016 < 0.05, it is unusual.

13. MC: For which of these scenarios is a binomial distribution not appropriate?

14. (A) Counting the number of heads in 10 coin flips.
15. (B) Counting the number of students who pass an exam out of 30, if each has a 70% chance.
16. (C) Counting the number of trials until a coin lands heads.
17. (D) Counting the number of defective items in a sample of 50 from a factory. Answer: (C) This is a geometric setting (trials until first success).

Last-Minute Cram Sheet

1. BINS: Binary, Independent, fixed n, Same p.
2. Mean = np; SD = ?[np(1?p)].
3. binompdf(n, p, k) = P(X=k); binomcdf(n, p, k) = P(X?k).
4. 10% condition: n-0.10N for independence without replacement. Always check!
5. Large Counts: np-10 and n(1?p)-10-normal approximation OK.
6. Normal approx: X-N(np, ?[np(1?p)]); z = (X-np)/?.
7. P(X-k) = 1-binomcdf(n, p, k?1).
8. Geometric-Binomial: Binomial = fixed n; geometric = trials until first success.
9. Label X and p in context (e.g., "X = number of defective bulbs; p = true defect rate").
10. Never use normal approx if np < 10 or n(1?p) < 10!