Binomial Probability Calculator
Binomial Probability Calculator
Calculate binomial distribution probabilities for exact, cumulative, and range outcomes with step-by-step solutions.
Binomial Probability Result
0.2461
Probability Analysis:
Binomial Distribution:
Probability Distribution Table:
Binomial distribution models the number of successes in a fixed number of independent trials.
What is Binomial Probability?
Binomial Probability is the probability of getting exactly k successes in n independent Bernoulli trials, where each trial has only two possible outcomes (success/failure) with constant probability p of success. The binomial distribution is one of the most important probability distributions in statistics, used to model binary outcomes in various fields including quality control, medicine, social sciences, and finance.
Binomial Distribution Formulas
Probability Mass Function
Exact probability
PMF formula
Cumulative Probability
At most k successes
CDF formula
Expected Value
Mean of distribution
Average successes
Variance
Spread of distribution
Dispersion measure
Key Formulas and Calculations
1. Binomial Probability Formula
Where:
C(n,k) = n! / (k! × (n-k)!) is the binomial coefficient
n = number of trials
k = number of successes
p = probability of success on each trial
2. Cumulative Probabilities
P(X ≥ k) = 1 - P(X ≤ k-1)
P(X < k) = P(X ≤ k-1)
P(X > k) = 1 - P(X ≤ k)
3. Distribution Properties
| Property | Formula | Interpretation | Example (n=10, p=0.5) |
|---|---|---|---|
| Mean (Expected Value) | μ = n × p | Average number of successes | 5.0 |
| Variance | σ² = n × p × (1-p) | Spread of distribution | 2.5 |
| Standard Deviation | σ = √[n × p × (1-p)] | Typical deviation from mean | 1.581 |
| Mode | ⌊(n+1)p⌋ or ⌊(n+1)p⌋-1 | Most likely number of successes | 5 |
Real-World Applications
Quality Control & Manufacturing
- Defect Analysis: Probability of defective items in a batch
- Acceptance Sampling: Determining sample sizes for quality checks
- Process Control: Monitoring production process stability
- Reliability Testing: Probability of component failures
Medicine & Healthcare
- Clinical Trials: Success rates of treatments
- Disease Prevalence: Probability of disease occurrence
- Drug Effectiveness: Response rates to medication
- Genetic Inheritance: Probability of genetic traits
Business & Finance
- Risk Management: Probability of loan defaults
- Marketing Campaigns: Response rates to promotions
- Sales Forecasting: Probability of sales conversions
- Customer Retention: Probability of customer churn
Social Sciences & Surveys
- Polling Accuracy: Probability of survey responses
- Voting Behavior: Probability of voter turnout
- Educational Testing: Probability of correct answers
- Social Research: Behavior pattern probabilities
Common Binomial Probability Examples
| Scenario | Trials (n) | Success Prob (p) | Successes (k) | Probability |
|---|---|---|---|---|
| Coin Toss (Heads) | 10 | 0.5 | 5 | 0.2461 |
| Dice Roll (Six) | 20 | 1/6 ≈ 0.1667 | 3 | 0.2379 |
| Quality Control (Defect) | 100 | 0.02 | 2 | 0.2734 |
| Survey Response (Yes) | 50 | 0.6 | 30 | 0.1146 |
| Basketball Free Throw | 10 | 0.7 | 8 | 0.2335 |
Step-by-Step Calculation Process
Example 1: Coin Toss (n=10, p=0.5, k=5)
- Identify parameters: n=10, p=0.5, k=5
- Calculate binomial coefficient: C(10,5) = 10!/(5!×5!) = 252
- Calculate probability of k successes: p^k = 0.5^5 = 0.03125
- Calculate probability of n-k failures: (1-p)^(n-k) = 0.5^5 = 0.03125
- Multiply: P(X=5) = 252 × 0.03125 × 0.03125 = 0.24609375
- Interpretation: Probability of exactly 5 heads in 10 tosses is 24.61%
Example 2: Cumulative Probability (n=10, p=0.5, k≤5)
- Calculate individual probabilities for k=0 to 5
- P(X=0) = C(10,0) × 0.5^0 × 0.5^10 = 0.0009766
- P(X=1) = C(10,1) × 0.5^1 × 0.5^9 = 0.0097656
- P(X=2) = C(10,2) × 0.5^2 × 0.5^8 = 0.0439453
- P(X=3) = C(10,3) × 0.5^3 × 0.5^7 = 0.1171875
- P(X=4) = C(10,4) × 0.5^4 × 0.5^6 = 0.2050781
- P(X=5) = C(10,5) × 0.5^5 × 0.5^5 = 0.2460938
- Sum: P(X≤5) = 0.0009766 + 0.0097656 + 0.0439453 + 0.1171875 + 0.2050781 + 0.2460938 = 0.6230469
- Interpretation: Probability of at most 5 heads in 10 tosses is 62.30%
Related Calculators
Frequently Asked Questions (FAQs)
Q: When can I use the binomial distribution?
A: Use binomial distribution when you have: 1) Fixed number of trials (n), 2) Independent trials, 3) Constant probability of success (p), 4) Only two possible outcomes (success/failure). If any condition fails, consider other distributions like hypergeometric (dependent trials) or Poisson (variable number of trials).
Q: What's the difference between binomial and normal distribution?
A: Binomial is discrete (counts of successes), while normal is continuous. However, when n is large and p is not extreme (np ≥ 5 and n(1-p) ≥ 5), binomial can be approximated by normal distribution with μ=np and σ=√[np(1-p)].
Q: How do I calculate cumulative probabilities?
A: Cumulative probabilities sum individual probabilities: P(X≤k) = Σ P(X=i) for i=0 to k. Use complement rule for "at least" probabilities: P(X≥k) = 1 - P(X≤k-1). Our calculator handles all cumulative probability types automatically.
Q: What if my probability p is greater than 1 or negative?
A: Probability p must be between 0 and 1 inclusive (0 ≤ p ≤ 1). Values outside this range are invalid because probability cannot exceed 100% or be negative. If p=0, success is impossible; if p=1, success is certain.
Master probability calculations with Toolivaa's free Binomial Probability Calculator, and explore more statistical tools in our Probability Calculators collection.