AP Statistics (AP Stats): Geometric Distributions (BITS conditions, mean = 1/p)

AP Statistics – Geometric Distributions (BITS conditions, mean = 1/p)

AP Statistics: Geometric Distributions – Exam-Ready Study Guide

What This Is

A geometric distribution models the number of trials needed to get the first success in repeated, independent Bernoulli trials (each with the same probability of success p). Unlike the binomial distribution (which counts the number of successes in n trials), the geometric distribution counts how many trials until the first success. This is essential on the AP exam for probability questions involving "waiting time" scenarios (e.g., how many free throws until the first miss, how many job applications until the first offer, how many defective items until the first failure).

Key Terms & Formulas

• Geometric Setting (BITS conditions):
• Binary: Each trial has two outcomes (success/failure).
• Independent: Trials do not affect each other.
• Trials: Count the number of trials until the first success.

• Same probability: Probability of success p is constant for each trial.

• Geometric Probability Formula: ( P(X = k) = (1 - p)^{k-1} \cdot p )

• X = number of trials until the first success.
• k = specific number of trials (e.g., k = 3 means first success on the 3rd trial).

• p = probability of success on a single trial.

• Mean (Expected Value) of a Geometric Distribution: ( \mu_X = \frac{1}{p} )

• Interpretation: On average, it takes ( \frac{1}{p} ) trials to get the first success.

• Standard Deviation of a Geometric Distribution: ( \sigma_X = \frac{\sqrt{1 - p}}{p} )

• Cumulative Geometric Probability (TI-84):

• geometcdf(p, k)-Probability of first success on or before the kth trial (i.e., P(X-k)).

• geometpdf(p, k)-Probability of first success exactly on the kth trial (i.e., P(X = k)).

• 10% Condition (for Independence): If sampling without replacement, the population must be at least 10 times larger than the sample to assume independence.

Step-by-Step / Process Flow

How to solve a geometric distribution FRQ:

1. Identify the Distribution
2. Ask: "Are we counting trials until the first success?" If yes, it’s geometric (not binomial).

3. Check BITS conditions (Binary, Independent, Trials until first success, Same p).

4. Define the Random Variable

5. Let X = number of trials until the first success.

6. State p (probability of success on a single trial).

7. Calculate Probabilities

8. For P(X = k): Use geometpdf(p, k) or the formula ( (1 - p)^{k-1} \cdot p ).
9. For P(X-k): Use geometcdf(p, k).

10. For P(X > k): ( 1 - P(X-k) = (1 - p)^k ).

11. Find the Mean (Expected Value)

12. ( \mu_X = \frac{1}{p} ). Interpret in context (e.g., "On average, it takes 5 trials to get the first success").

13. Answer the Question in Context

14. Avoid generic answers. Example: "There is a 12.8% chance the first defective item is found on the 4th inspection."

Common Mistakes

• Mistake: Confusing geometric and binomial distributions.

• Correction: Binomial counts number of successes in n trials; geometric counts trials until the first success. Example: "Number of free throws until the first miss" = geometric. "Number of misses in 10 free throws" = binomial.

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

• Correction: If the problem involves sampling without replacement (e.g., drawing cards from a deck), check that the population is ? 10 × sample size to assume independence.

• Mistake: Misusing geometcdf vs. geometpdf.

• Correction:

  • geometpdf(p, k) = P(X = k) (exactly k trials).
  • geometcdf(p, k) = P(X-k) (first success on or before k trials).

• Mistake: Incorrectly calculating P(X > k).

• Correction: P(X > k) = ( (1 - p)^k ) (no success in the first k trials). Do not use 1 - geometcdf(p, k) (this gives P(X-k+1)).

• Mistake: Misinterpreting the mean.

• Correction: The mean ( \frac{1}{p} ) is the expected number of trials until the first success, not the probability.

AP Exam Insights

• Frequent FRQ Setups:
• "How many trials until the first success?" (e.g., free throws, defective items, job offers).
• "Find the probability the first success occurs on the kth trial."
• "What is the expected number of trials until the first success?"

• "Find the probability it takes more than k trials to get the first success."

• Tricky Distinctions:

• Geometric vs. Binomial: The AP exam loves testing whether a scenario is geometric or binomial. Key phrase for geometric: "until the first success."

• Calculator Pitfalls: Students often mix up geometpdf and geometcdf. Remember: pdf = exact, cdf = cumulative.

• Common Calculator Errors:

• Forgetting to define p correctly (e.g., if p = 0.2, use geometpdf(0.2, 3), not geometpdf(3, 0.2)).

• Using binompdf/binomcdf instead of geometric functions.

• Real-World Contexts:

• Quality control (defective items).
• Sports (free throws, hits in baseball).
• Customer service (calls until first complaint).

Quick Check Questions

1. Multiple Choice: A basketball player makes 80% of her free throws. What is the probability she makes her first miss on the 5th attempt?
2. (A) ( 0.8^4 \cdot 0.2 )
3. (B) ( 0.8^5 )
4. (C) ( 0.2^4 \cdot 0.8 )
5. (D) ( 1 - 0.8^5 )

Answer: (A) ( 0.8^4 \cdot 0.2 ) Explanation: This is P(X = 5) for a geometric distribution with p = 0.2 (probability of a miss).

1. FRQ Part: A factory produces light bulbs with a 2% defect rate. Assume independence.
2. (a) What is the probability the first defective bulb is found on the 10th inspection?
3. (b) What is the expected number of bulbs inspected until the first defect is found?

Answer: - (a) ( P(X = 10) = (0.98)^9 \cdot 0.02 ) or geometpdf(0.02, 10)-0.0167. - (b) ( \mu_X = \frac{1}{0.02} = 50 ) bulbs.

1. Multiple Choice: For a geometric distribution with p = 0.1, what is P(X > 3)?
2. (A) 0.729
3. (B) 0.001
4. (C) 0.9
5. (D) 0.271

Answer: (A) 0.729 Explanation: P(X > 3) = ( (1 - 0.1)^3 = 0.9^3 = 0.729 ).

Last-Minute Cram Sheet

1. Geometric = "Trials until first success" (vs. binomial = "successes in n trials").
2. BITS conditions: Binary, Independent, Trials until first success, Same p.
3. Formula: ( P(X = k) = (1 - p)^{k-1} \cdot p ).
4. Mean: ( \mu_X = \frac{1}{p} ) (interpret as "expected trials until first success").
5. TI-84: geometpdf(p, k) = P(X = k), geometcdf(p, k) = P(X-k).
6. P(X > k) = ( (1 - p)^k ) (no success in first k trials).
7. Check 10% condition if sampling without replacement.
8. Don’t confuse geometpdf and geometcdf!
9. p is the probability of success, not failure.
10. Always interpret answers in context (e.g., "There is a 12% chance...").