Geometric Distribution
Print PageA Geometric Distribution Calculator (also known as a Geometric Probability Calculator, First Success Trial Utility, Cumulative Geometric CDF Analyzer, or Bernoulli Waiting Time Calculator) computes exact single-point probabilities (P(X = k) = (1 - p)k-1 · p), lower cumulative probabilities (P(X ≤ k) = 1 - (1 - p)k), upper cumulative probabilities (P(X > k) = (1 - p)k), expected mean (μ = 1 ÷ p), variance (σ2 = (1 - p) ÷ p2), and standard deviation for the trial number k of the first success in independent Bernoulli trials with constant success probability p.
In sales cold calling, software bug detection, and medical treatment trials, the geometric distribution models the number of attempts required to achieve the very first success, possessing the discrete memoryless property.
Our free online Geometric Distribution Calculator provides instant calculations across all trial parameters:
- Probability Mass Function (PMF / 1st Success on Trial k):
P(X = k) = (1 - p)k-1 · p(fork = 1, 2, 3...). - Lower Cumulative Distribution Function (CDF / P[X ≤ k]):
P(X ≤ k) = 1 - (1 - p)k. - Upper Cumulative Probability (P[X > k]):
P(X > k) = (1 - p)k. - Expected Mean Trials to 1st Success (μ):
μ = 1 ÷ p. - Variance (σ2):
σ2 = (1 - p) ÷ p2. - Standard Deviation (σ):
σ = √[ 1 - p ] ÷ p. - Discrete Memoryless Property Formula:
P(X > s + t | X > s) = P(X > t) = (1 - p)t.
Master Geometric Probability Reference Table (p = 0.10 Sales Lead Conversion)
The table below displays single-point probabilities P(X = k), lower cumulative probabilities P(X ≤ k), upper cumulative probabilities P(X > k), and expected values for a cold calling campaign with p = 0.10 (10% conversion rate):
| Trial Number (k Call Attempt) | PMF Formula (1 – p)k-1 · p | Exact P(X = k) | Lower Cumulative P(X ≤ k) | Upper Cumulative P(X > k) | Odds Ratio Representation |
|---|---|---|---|---|---|
| k = 1st Call | (0.90)0 · 0.10 | 10.0000% (0.1000) | 10.0000% | 90.0000% | 1 in 10 Calls |
| k = 2nd Call | (0.90)1 · 0.10 | 9.0000% (0.0900) | 19.0000% | 81.0000% | 1 in 11.1 Calls |
| k = 3rd Call | (0.90)2 · 0.10 | 8.1000% (0.0810) | 27.1000% | 72.9000% | 1 in 12.3 Calls |
| k = 5th Call (5-Call Benchmark) | (0.90)4 · 0.10 | 6.5610% (6.56%) | 40.9510% (40.95%) | 59.0490% (59.05%) | 1 in 15.2 Calls |
| k = 10th Call (μ Expected Mean) | (0.90)9 · 0.10 | 3.8742% (3.87%) | 65.1322% (65.13%) | 34.8678% (34.87%) | 1 in 25.8 Calls |
Step-by-Step Sales Lead Conversion Calculation (p = 0.10)
To evaluate a sales representative cold calling prospects with a baseline success probability of p = 0.10 (10% closed sale rate):
Step 1 (Calculate Expected Mean Calls to 1st Sale μ): μ = 1 ÷ p = 1 ÷ 0.10 = 10.0 calls
Step 2 (Calculate Variance σ^2): σ^2 = (1 - 0.10) ÷ (0.10)^2 = 0.90 ÷ 0.01 = 90.0 &implies; σ = √90.0 = 9.487 calls
Step 3 (Calculate Probability 1st Sale Occurs on Exact 5th Call P[X = 5]): P(X = 5) = (0.90)^4 · 0.10 = 0.6561 · 0.10 = 0.06561 ≈ 6.56%
Step 4 (Calculate Probability 1st Sale Occurs Within 5 Calls P[X ≤ 5]): P(X ≤ 5) = 1 - (0.90)^5 = 1 - 0.59049 = 0.40951 ≈ 40.95%
Thus, the sales rep expects to make 10 calls to close their first sale, with a 40.95% probability of closing a sale within the first 5 calls.
Discrete Probability Models: Geometric vs. Binomial vs. Negative Binomial
Below is a comparative reference chart detailing when to use the Geometric distribution versus related discrete models:
| Probability Distribution | Target Variable Being Calculated | Fixed Input Condition | Primary Practical Application |
|---|---|---|---|
| Geometric Distribution | Trial number k of the VERY FIRST success (r = 1) | Fixed r = 1 success | Sales cold calling, component first-failure times. |
| Binomial Distribution | Number of successes k in a fixed series of n trials | Fixed n trials | Quality control defects per 100-unit batch. |
| Negative Binomial Distribution | Trial number k of the r-th success (Generalizes Geometric) | Fixed r successes (r ≥ 1) | Closing 5 sales contracts or 3 customer deals. |
History & Mathematics: 1713 Jacob Bernoulli to 1837 Siméon Poisson
1713 Jacob Bernoulli & Bernoulli Trials
In 1713, Swiss mathematician Jacob Bernoulli formulated the mathematical foundations of independent binary trial sequences in Ars Conjectandi.
1837 Siméon Denis Poisson & Discrete Waiting Times
In 1837, French mathematician Siméon Denis Poisson formalized discrete waiting time limits, establishing the mathematical relationship between Geometric and Negative Binomial probability distributions.
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Frequently Asked Questions (FAQ)
What is the Geometric Distribution formula?
The single-point probability formula is P(X = k) = (1 - p)k-1 · p.
What is the expected mean of a Geometric distribution?
The expected mean number of trials until the first success is μ = 1 ÷ p.
What does the Memoryless Property mean for the Geometric distribution?
It means that failed previous trials do not increase or decrease the probability of success on future trials (P(X > s + t | X > s) = P(X > t)).