Geometric Distribution

When to Use Geometric

Geometric distribution: Models number of trials until first success

Conditions:

1. Binary: Each trial has two outcomes (success/failure)
2. Independent: Trials independent
3. Same: Probability of success (p) constant
4. Trials until success: Count trials up to and including first success

Key difference from Binomial: Number of trials NOT fixed (varies randomly)

Notation: X ~ Geometric(p)

Geometric Probability Formula

Probability first success occurs on trial k:

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

Interpretation:

• Fail (k-1) times: probability (1-p)^(k-1)
• Succeed on kth trial: probability p
• Multiply together

Example 1: Rolling a Six

Roll die until get a 6. Find P(first 6 on roll 3).

p = 1/6 (probability of rolling 6)

$P(X = 3) = (5/6)^2 \cdot (1/6) = (25/36)(1/6) = 25/216 \approx 0.116$

Interpretation: Fail twice, succeed third time

Example 2: Free Throws

Player makes 80% of free throws. Find P(first make on shot 4).

p = 0.8

$P(X = 4) = (0.2)^3 (0.8) = 0.008(0.8) = 0.0064$

Interpretation: Miss first 3 (unlikely for 80% shooter!), make 4th

Mean and Standard Deviation

Mean (Expected Value):

$\mu_X = \frac{1}{p}$

Standard Deviation:

$\sigma_X = \frac{\sqrt{1-p}}{p}$

Example: Roll die until 6 (p = 1/6)

$\mu_X = \frac{1}{1/6} = 6$

$\sigma_X = \frac{\sqrt{5/6}}{1/6} = \sqrt{5/6} \cdot 6 = 6\sqrt{5/6} \approx 5.48$

Interpretation: On average, takes 6 rolls to get first 6

Cumulative Probabilities

P(X ≤ k): First success within k trials

$P(X \leq k) = 1 - (1-p)^k$

Logic: Complement of "fail all k trials"

Example: Die rolling, P(first 6 within 3 rolls)

$P(X \leq 3) = 1 - (5/6)^3 = 1 - 125/216 = 91/216 \approx 0.421$

P(X > k): More than k trials needed

$P(X > k) = (1-p)^k$

Example: P(need more than 3 rolls for first 6)

$P(X > 3) = (5/6)^3 = 125/216 \approx 0.579$

Calculator Commands (TI-83/84)

geometpdf(p, k): P(X = k)

• Example: geometpdf(1/6, 3)

geometcdf(p, k): P(X ≤ k)

• Example: geometcdf(1/6, 3)

Access: 2nd VARS (DISTR) → geometpdf or geometcdf

Geometric vs Binomial

Binomial:

• Fixed n trials
• Count successes (X can be 0 to n)
• Question: "How many successes in n trials?"

Geometric:

• Variable trials (until first success)
• X = trial number of first success (1, 2, 3, ...)
• Question: "How many trials until first success?"

Example distinguishing:

• "Flip 10 coins, count heads" → Binomial
• "Flip until first heads" → Geometric

Memoryless Property

Unique property of geometric distribution:

$P(X > a + b | X > a) = P(X > b)$

Interpretation: If already waited a trials without success, probability of waiting b more trials same as starting fresh

Example: Rolling die

• P(wait more than 6 rolls | already waited 3) = P(wait more than 3 rolls)
• Past failures don't affect future (each roll independent)

Applications

Manufacturing: Inspecting items until find defect
Quality control: Testing until failure
Gaming: Playing until win
Biology: Trials until mutation occurs
Sports: At-bats until hit

Example 3: Quality Control

2% of widgets are defective. Inspect widgets one at a time.

(a) Expected number inspected until find defect?

$\mu = \frac{1}{p} = \frac{1}{0.02} = 50$

Expect to inspect 50 widgets on average

(b) P(find defect within 10 inspections)?

$P(X \leq 10) = 1 - (0.98)^{10} = 1 - 0.817 = 0.183$

(c) P(need more than 100 inspections)?

$P(X > 100) = (0.98)^{100} \approx 0.133$

Probability Distribution Graph

For Geometric(0.3):

• Always right-skewed (starts at X=1)
• Decreasing probabilities (most likely: X=1)
• Long right tail (theoretically infinite)

For Geometric(0.8):

• Strongly concentrated at X=1
• Rapid decrease (high p means quick success likely)

Relationship to Binomial

For small p: Geometric and binomial related

If X ~ Geometric(p), then after n trials, number of successes ~ Binomial(n, p)

But they answer different questions!

Common Mistakes

❌ Using geometric when trials are fixed (should use binomial)
❌ Starting count at 0 instead of 1 (X=1 is first trial)
❌ Confusing P(X = k) with P(X ≤ k)
❌ Forgetting memoryless property
❌ Wrong mean formula (it's 1/p, not p)

Practice Strategy

1. Identify: Trials until first success? (→ Geometric)
2. Find p: Probability of success each trial
3. Determine question: Exactly k trials? At most k? More than k?
4. Apply formula or use calculator
5. Interpret: Does answer make sense?

Quick Reference

Use when: Counting trials until first success

Probability: $P(X = k) = (1-p)^{k-1} p$

Mean: $\mu = \frac{1}{p}$

SD: $\sigma = \frac{\sqrt{1-p}}{p}$

Cumulative: $P(X \leq k) = 1 - (1-p)^k$

Calculator:

• geometpdf(p, k) for P(X = k)
• geometcdf(p, k) for P(X ≤ k)

Remember: Geometric counts trials until first success. Mean = 1/p makes sense: if p=0.5, expect success on trial 2 on average!