Binomial Probability Distribution Calculator
Calculate exact and cumulative probabilities for binomial experiments. Find the chance of getting a specific number of successes in a fixed number of independent trials.
Binomial Distribution Calculator
Calculate probabilities for binomial distributions
About Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success.
Requirements: Fixed number of trials (n), two possible outcomes (success/failure), constant probability (p), and independent trials.
Formula: P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
How the Binomial Distribution Calculator Works
Enter the number of trials (n) - the total number of independent attempts. Input the probability of success (p) for each trial as a decimal between 0 and 1. Specify the number of successes (x) you're interested in.
Select the probability type: exactly x successes, at most x successes (cumulative up to x), at least x successes (cumulative from x), or more than x successes. The calculator uses the binomial formula: P(X=x) = C(n,x) × p^x × (1-p)^(n-x).
Results display the probability as decimal and percentage. The distribution graph shows probabilities for all possible outcomes from 0 to n successes. Mean (np) and standard deviation (√(np(1-p))) help interpret the distribution's center and spread.
When You'd Actually Use This
Quality control sampling
Find probability of finding exactly 2 defective items in a sample of 20, when defect rate is 5%. Helps set acceptable quality levels.
Medical treatment success rates
Calculate probability that at least 8 out of 10 patients respond to treatment with 70% success rate. Useful for setting realistic expectations.
Survey response analysis
Determine likelihood of getting at least 50 "yes" responses from 100 people when true proportion is 40%. Helps assess survey reliability.
Sports performance prediction
Find probability a basketball player makes exactly 7 of 10 free throws given 75% career average. Compare actual performance to expected.
Genetics inheritance patterns
Calculate probability of exactly 3 children inheriting a dominant trait from heterozygous parents (75% chance each). Applies Mendelian genetics.
Marketing campaign planning
Estimate probability of getting at least 100 conversions from 1000 emails with 8% historical conversion rate. Helps set campaign goals.
What to Know Before Using
Trials must be independent.Each trial's outcome doesn't affect others. Coin flips are independent. Drawing cards without replacement is not - use hypergeometric instead.
Success probability must be constant.The probability p stays the same for every trial. If probability changes (like learning effects), binomial doesn't apply.
Only two outcomes per trial.Each trial results in success or failure. For more than two outcomes, use multinomial distribution instead.
Normal approximation works for large n.When np ≥ 10 and n(1-p) ≥ 10, binomial approximates normal distribution. Useful for quick estimates with large samples.
Pro tip: "At least" and "at most" calculations can be computationally intensive for large n. Use complement rule: P(at least 5) = 1 - P(at most 4). Often faster to calculate the smaller tail.
Common Questions
What's the expected number of successes?
Expected value = n × p. If you flip 100 coins (p=0.5), expect 50 heads. This is the long-run average over many repetitions.
How do I calculate combinations C(n,x)?
C(n,x) = n! / (x! × (n-x)!). It counts ways to choose x items from n. Also written as "n choose x" or with subscript notation.
When is the distribution symmetric?
When p = 0.5, the distribution is symmetric. When p < 0.5, it skews right. When p > 0.5, it skews left.
What's the most likely outcome?
The mode is floor((n+1) × p). For n=10, p=0.3, mode is floor(11 × 0.3) = 3. This is the single most probable number of successes.
Can probability exceed 1?
No. Individual probabilities range from 0 to 1. The sum of all probabilities from 0 to n successes equals exactly 1.
What if n is very large?
Use normal approximation with mean = np and SD = √(np(1-p)). Apply continuity correction by adding/subtracting 0.5 for better accuracy.
How does this differ from Poisson?
Binomial has fixed n trials. Poisson models counts over time/space with no fixed maximum. Use Poisson when n is large and p is small.
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