Binomial Distribution Calculator
Calculate binomial probabilities, cumulative distribution, mean, and standard deviation. Find the probability of k successes in n trials.
Probability
0.246094
Mean (ฮผ)
5.0000
Variance (ฯยฒ)
2.5000
Std. Deviation (ฯ)
1.5811
Distribution Parameters
n: 10 trials
p: 0.5 (success probability)
q: 0.5000 (failure probability)
Mode: 5
Probability Distribution
| k | P(X = k) | P(X โค k) | Bar |
|---|---|---|---|
| 0 | 0.0010 | 0.0010 | |
| 1 | 0.0098 | 0.0107 | |
| 2 | 0.0439 | 0.0547 | |
| 3 | 0.1172 | 0.1719 | |
| 4 | 0.2051 | 0.3770 | |
| 5 | 0.2461 | 0.6230 | |
| 6 | 0.2051 | 0.8281 | |
| 7 | 0.1172 | 0.9453 | |
| 8 | 0.0439 | 0.9893 | |
| 9 | 0.0098 | 0.9990 | |
| 10 | 0.0010 | 1.0000 |
Formula Used
P(X = k) = C(n,k) ร p^k ร (1-p)^(n-k)
Where C(n,k) = n! / (k! ร (n-k)!)
Related Calculators
About This Calculator
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. From coin flips to quality control, it's one of the most important probability distributions in statistics.
What is the Binomial Distribution? When you repeat an experiment n times, with each trial having probability p of success, the binomial distribution tells you the probability of getting exactly k successes. It answers questions like "What's the probability of getting exactly 7 heads in 10 coin flips?"
Requirements for Binomial Distribution:
- Fixed number of trials (n)
- Each trial is independent
- Only two outcomes: success or failure
- Constant probability of success (p)
Key Parameters:
- n: Number of trials
- p: Probability of success per trial
- k: Number of successes to find probability for
- Mean: ฮผ = np
- Variance: ฯยฒ = np(1-p)
This calculator computes probabilities, cumulative distributions, and distribution statistics. For related calculations, see our Combination Calculator and Normal Distribution Calculator.
How to Use the Binomial Distribution Calculator
- 1Enter the total number of trials (n).
- 2Enter the probability of success on each trial (p), between 0 and 1.
- 3Select the type of probability you want to calculate.
- 4Enter the number of successes (k) for exact or cumulative probabilities.
- 5For range probabilities, enter both start and end values.
- 6Review the calculated probability.
- 7Check the mean, variance, and standard deviation.
- 8Examine the probability distribution table.
- 9Visual bars show relative probabilities for each k.
- 10Use the formula reference to verify calculations.
The Binomial Formula
Understanding the mathematics behind binomial probability.
Probability Mass Function (PMF)
P(X = k) = C(n,k) ร p^k ร (1-p)^(n-k)
Where:
- C(n,k) = n! / (k!(n-k)!) is the binomial coefficient
- p = probability of success
- (1-p) = q = probability of failure
- n = number of trials
- k = number of successes
Breaking Down the Formula
C(n,k): Number of ways to choose k successes from n trials p^k: Probability of k successes occurring (1-p)^(n-k): Probability of (n-k) failures occurring
Example: 3 Heads in 5 Coin Flips
n = 5, k = 3, p = 0.5
P(X = 3) = C(5,3) ร 0.5ยณ ร 0.5ยฒ = 10 ร 0.125 ร 0.25 = 0.3125 (31.25%)
Cumulative Distribution Function (CDF)
P(X โค k) = ฮฃแตขโโแต P(X = i)
Sum of all probabilities from 0 to k.
Mean, Variance, and Standard Deviation
Key statistics that describe the distribution.
Expected Value (Mean)
ฮผ = E[X] = np
The average number of successes expected.
Example: Flip coin 100 times (n=100, p=0.5) Expected heads: ฮผ = 100 ร 0.5 = 50
Variance
ฯยฒ = Var(X) = np(1-p) = npq
Measure of spread around the mean.
Maximum variance occurs at p = 0.5.
Standard Deviation
ฯ = โ(npq)
Same units as the random variable.
Example Calculations
For n = 100, p = 0.3:
- Mean: ฮผ = 100 ร 0.3 = 30
- Variance: ฯยฒ = 100 ร 0.3 ร 0.7 = 21
- Std Dev: ฯ = โ21 โ 4.58
68-95-99.7 Rule Approximation
For large n, binomial โ normal:
- About 68% of values within ฮผ ยฑ ฯ
- About 95% within ฮผ ยฑ 2ฯ
- About 99.7% within ฮผ ยฑ 3ฯ
Common Applications
Real-world situations that follow binomial distribution.
Quality Control
Scenario: A factory produces items with 2% defect rate. In a batch of 50 items:
- P(0 defects) = 0.364
- P(โค 2 defects) = 0.922
- Expected defects = 50 ร 0.02 = 1
Medical Testing
Scenario: Drug has 70% effectiveness. For 10 patients:
- P(exactly 7 respond) = 0.267
- P(at least 8 respond) = 0.383
- Expected responders = 7
Survey Sampling
Scenario: 40% of population prefers Brand A. In sample of 25:
- P(exactly 10 prefer A) = 0.161
- Expected preference = 10
- ฯ = โ(25 ร 0.4 ร 0.6) = 2.45
Sports and Games
Scenario: Basketball player has 80% free throw rate. For 5 free throws:
- P(all 5 made) = 0.328
- P(at least 4 made) = 0.737
- Expected makes = 4
A/B Testing
Scenario: Website has 5% conversion rate. For 200 visitors:
- Expected conversions = 10
- ฯ = โ(200 ร 0.05 ร 0.95) = 3.08
- P(15+ conversions) helps detect improvements
Cumulative Probabilities
Calculating "at least" and "at most" probabilities.
Types of Questions
Exact: P(X = k) "Exactly k successes"
At Most: P(X โค k) "k or fewer successes" (CDF)
At Least: P(X โฅ k) = 1 - P(X โค k-1) "k or more successes"
Between: P(a โค X โค b) = P(X โค b) - P(X โค a-1) "Between a and b successes inclusive"
Example: Pass/Fail Exam
n = 20 questions, guess randomly (p = 0.25), need 12 to pass
P(pass by guessing) = P(X โฅ 12) = 1 - P(X โค 11) = 1 - 0.9991 = 0.0009 (0.09%)
Complement Rule
P(X โฅ k) = 1 - P(X โค k-1)
Often easier to calculate the complement.
Tables and Technology
For large n, use:
- Binomial tables
- Calculator/software
- Normal approximation
Normal Approximation to Binomial
When n is large, binomial approaches normal distribution.
Rule of Thumb
Use normal approximation when:
- np โฅ 10 AND n(1-p) โฅ 10
Some texts use np โฅ 5 and nq โฅ 5.
The Approximation
If X ~ Binomial(n, p), then approximately:
X ~ Normal(ฮผ = np, ฯ = โ(npq))
Continuity Correction
When using normal to approximate binomial:
| Binomial | Normal |
|---|---|
| P(X = k) | P(k - 0.5 < X < k + 0.5) |
| P(X โค k) | P(X < k + 0.5) |
| P(X โฅ k) | P(X > k - 0.5) |
| P(X < k) | P(X < k - 0.5) |
| P(X > k) | P(X > k + 0.5) |
Example
n = 100, p = 0.4 Find P(X โค 45)
Normal approximation:
- ฮผ = 100 ร 0.4 = 40
- ฯ = โ(100 ร 0.4 ร 0.6) = 4.90
- z = (45.5 - 40) / 4.90 = 1.12
- P(Z โค 1.12) โ 0.869
Exact binomial: P(X โค 45) = 0.872
Very close!
Related Distributions
How binomial connects to other probability distributions.
Bernoulli Distribution
A binomial with n = 1.
P(X = 1) = p, P(X = 0) = 1-p
Binomial = sum of n Bernoulli trials.
Geometric Distribution
Number of trials until first success.
P(X = k) = (1-p)^(k-1) ร p
Different question: "How many tries until success?"
Negative Binomial
Number of trials until r successes.
Generalizes geometric (r = 1).
Poisson Distribution
Limiting case of binomial:
- When n โ โ and p โ 0
- With np = ฮป constant
P(X = k) = e^(-ฮป) ร ฮป^k / k!
Hypergeometric Distribution
Sampling without replacement.
Use when:
- Drawing from finite population
- Success probability changes with each draw
When to Use Each
| Situation | Distribution |
|---|---|
| Fixed n, with replacement | Binomial |
| Fixed n, without replacement | Hypergeometric |
| Trials until first success | Geometric |
| Trials until r successes | Negative Binomial |
| Rare events in interval | Poisson |
Pro Tips
- ๐กCheck all four conditions: fixed n, independence, binary outcomes, constant p.
- ๐กMean = np gives you the expected number of successes.
- ๐กUse complement rule: P(X โฅ k) = 1 - P(X โค k-1) is often easier.
- ๐กNormal approximation works when np โฅ 10 and n(1-p) โฅ 10.
- ๐กApply continuity correction when using normal approximation.
- ๐กMaximum variance occurs at p = 0.5.
- ๐กThe distribution is symmetric only when p = 0.5.
- ๐กFor rare events (small p, large n), consider Poisson approximation.
- ๐กSampling without replacement needs hypergeometric, not binomial.
- ๐กTables and calculators are essential for large n values.
- ๐กThe mode is approximately np for most cases.
- ๐กStandard deviation = โ(npq) measures typical variation.
Frequently Asked Questions
Use binomial when: (1) you have a fixed number of trials, (2) each trial is independent, (3) each trial has only two outcomes (success/failure), and (4) the probability of success is constant. Classic examples: coin flips, quality control sampling, yes/no surveys.
