🎲 Geometric Distribution

When to Use Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials.

Key Difference from Binomial:

• Binomial: Fixed number of trials, count successes (X = # successes in n trials)
• Geometric: Fixed target (1 success), count trials until we get it

Conditions for Geometric Distribution (BIS minus Fixed n)

1. Bernoulli trials: each trial has two outcomes (success/failure)
2. Independence: trials are independent
3. Same probability: probability p of success is constant on each trial
4. Fixed n: NOT fixed — we stop when we get first success

Probability Formula

For X = number of trials until first success:

$P(X = k) = (1 - p)^{k-1} \cdot p$

where:

• $k$ = 1, 2, 3, ... (the trial number on which first success occurs)
• $p$ = probability of success on each trial
• $(1 - p)^{k-1}$ = (k−1) failures before the first success

Mean and Standard Deviation

$E(X) = \frac{1}{p}$

$SD(X) = \sqrt{\frac{1-p}{p^2}} = \frac{\sqrt{1-p}}{p}$

Worked Example

A basketball player makes 60% of free throws. What is the probability that her first made free throw occurs on the 3rd attempt?

Given: p = 0.60, k = 3

Calculation: $P(X = 3) = (0.40)^{3-1} \cdot (0.60) = (0.40)^2 \cdot 0.60 = 0.16 \cdot 0.60 = 0.096$

Interpretation: There is a 9.6% chance her first made free throw occurs on the 3rd attempt (meaning she misses the first two and makes the third).

Mean: $E(X) = \frac{1}{0.60} \approx 1.67$ attempts on average

Common Mistakes

1. Using k instead of (k−1): Always use $(1-p)^{k-1}$, not $(1-p)^k$
2. Confusing with binomial: "First success on trial k" is geometric, "k successes in n trials" is binomial
3. Forgetting that k starts at 1: On the first trial (k=1), $P(X=1) = p$ (immediate success)
4. Not recognizing when to use it: Look for phrases like "first," "until success," "wait for"

Decision Rule / When to Apply

Use geometric distribution when:

• Asked for probability of first success occurring on trial k
• Asked for expected number of trials until first success
• Sampling until you find one item with a desired property

AP Exam Tip

Geometric problems often appear in free-response questions about quality control, customer service (first complaint), or medical applications (first positive test). Watch for cumulative probability questions: $P(X \leq k)$ requires summing individual probabilities or recognizing the cumulative geometric formula.