Binomial Probability Distribution Calculator
Calculate probability of k successes in n independent trials with step-by-step solutions
P(X = k) = C(n, k) × pk × (1-p)(n-k)
Probability of exactly k successes in n independent trials
Positive integer (1-1000)
Must satisfy 0 ≤ k ≤ n
Value between 0 and 1
Quick Examples
Invalid Input
P(X = ) with n = , p =
P(X ≤ ) with n = , p =
Distribution Statistics for n = , p =
Expected Value (μ)
Variance (σ²)
Std Deviation (σ)
Expected Value (μ = np)
Variance (σ² = np(1-p))
Std Deviation (σ)
Binomial Coefficient
C(, ) =
Number of ways to choose successes from trials
Step-by-Step Solution
Identify values
n = , k = , p =
Calculate binomial coefficient
C(, ) = ! / (! × !) =
Apply the formula
P(X = ) = × ×
Final result
P(X = ) = ≈
Probability Distribution
Showing first 21 values (k = 0 to 20)
Common Binomial Scenarios
| Scenario | n | p | Expected (μ) |
|---|---|---|---|
| Fair coin (10 flips) | 10 | 0.5 | 5 |
| Die rolls (getting 6) | 12 | 0.167 | 2 |
| Quality control (1% defect) | 100 | 0.01 | 1 |
| Free throws (80% shooter) | 10 | 0.8 | 8 |
| Survey response (30%) | 50 | 0.3 | 15 |
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About Binomial Probability Distribution Calculator
What is the Binomial Probability Distribution?
The binomial probability distribution describes the probability of obtaining exactly k successes in n independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant.
The Binomial Probability Formula
P(X = k) = C(n, k) × p^k × (1-p)^(n-k)
Where:
- P(X = k) = Probability of exactly k successes
- n = Number of trials
- k = Number of successes desired
- p = Probability of success on a single trial
- (1-p) = Probability of failure (often denoted as q)
- C(n, k) = Binomial coefficient = n! / (k!(n-k)!)
Conditions for a Binomial Experiment
For a random variable X to follow a binomial distribution:
- Fixed number of trials (n) - The experiment consists of n identical trials
- Two possible outcomes - Each trial results in success or failure
- Independent trials - The outcome of one trial doesn't affect others
- Constant probability (p) - The probability of success is the same for each trial
Key Formulas
Expected Value (Mean)
μ = E(X) = n × p
The average number of successes you expect in n trials.
Variance
σ² = Var(X) = n × p × (1-p)
Standard Deviation
σ = √(n × p × (1-p))
Cumulative Probability
P(X ≤ k) = Σ P(X = i) for i = 0 to k
The probability of getting at most k successes.
Common Applications
1. Quality Control
- Defective products in a batch
- Pass/fail testing
2. Medical Research
- Drug effectiveness trials
- Success rates of treatments
3. Games and Gambling
- Coin flip outcomes
- Dice roll successes
- Lottery analysis
4. Sports Analytics
- Free throw success rates
- Batting averages
5. Surveys and Polling
- Yes/no response rates
- Election predictions
Example Calculations
Example 1: Coin Flips
Flipping a fair coin 10 times, probability of exactly 6 heads:
- n = 10, k = 6, p = 0.5
- P(X=6) = C(10,6) × 0.5^6 × 0.5^4 = 210 × 0.015625 × 0.0625 ≈ 0.2051
Example 2: Quality Control
If 5% of products are defective, probability of finding exactly 2 defective items in a sample of 20:
- n = 20, k = 2, p = 0.05
- P(X=2) = C(20,2) × 0.05^2 × 0.95^18 ≈ 0.1887
Important Notes
- When n is large and p is small, the binomial distribution can be approximated by the Poisson distribution
- When n is large, the binomial distribution approaches the normal distribution (Central Limit Theorem)
- The binomial coefficient C(n,k) can become very large, so use computational tools for large n values