Geometric Distribution Table

Geometric Distribution Table

The geometric distribution models the number of trials needed to get the first success, or equivalently, the number of failures before the first success.

Two Versions of Geometric Distribution

Version 1: X = number of trials until first success (X ≥ 1)

• P(X = k) = p(1-p)^(k-1)

Version 2: Y = number of failures before first success (Y ≥ 0)

• P(Y = k) = p(1-p)^k

This table uses Version 1 (trials until success).

Individual Probabilities P(X = k)

Number of Trials Until First Success

Cumulative Probabilities P(X ≤ k)

Probability of Success Within k Trials

Survival Function P(X > k)

Probability of Needing More Than k Trials

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

Mean, Variance, and Key Properties

Expected Number of Trials

Example Problems

Example 1: First Success

Problem: A basketball player has a 30% free throw percentage. What’s the probability they make their first shot on the 3rd attempt?

Solution (p = 0.30, k = 3):

• P(X = 3) = 0.1470 (from table)

Example 2: Within k Trials

Problem: What’s the probability of success within 5 trials?

Solution:

• P(X ≤ 5) = 0.8319 (from cumulative table)

Example 3: More Than k Trials

Problem: What’s the probability of needing more than 5 trials?

Solution:

• P(X > 5) = 1 - P(X ≤ 5) = 1 - 0.8319 = 0.1681

Memoryless Property

The geometric distribution is memoryless:

P(X > m + n | X > m) = P(X > n)

This means: given that you’ve failed m times, the probability of needing more than n additional trials equals the probability of needing more than n trials from the start.

Relationship to Other Distributions

When to Use Geometric Distribution

✓ Counting trials until first success
✓ Independent Bernoulli trials
✓ Constant probability p
✓ Waiting time problems

Related Resources

• Geometric Distribution Lesson - Complete guide
• Negative Binomial Table - Multiple successes
• Binomial Table - Fixed number of trials
• Probability Calculator - Calculate probabilities