Binomial Probability Calculator
Calculate the probability of exactly k successes in n independent trials, each with the same probability p of success.
P(X = k) — Exactly k successes
P(X ≤ k) — At most k successes
P(X ≥ k) — At least k successes
P(X < k) — Fewer than k
P(X > k) — More than k
Distribution Properties
Mean (μ)
Variance (σ²)
Std Deviation (σ)
Skewness
Probability Distribution
| k | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|
Binomial Probability Formula
Where:
- n = number of independent trials
- k = number of successes (the value you want to find the probability for)
- p = probability of success on a single trial
- C(n, k) = the binomial coefficient, also written as "n choose k" = n! / (k! · (n − k)!)
What Is a 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. It is one of the most widely used discrete probability distributions in statistics and is the foundation for many real-world probability problems.
Conditions for a Binomial Experiment
- Fixed number of trials (n): The experiment consists of a predetermined number of identical trials.
- Two outcomes: Each trial results in either a "success" or a "failure."
- Constant probability (p): The probability of success is the same for every trial.
- Independence: The outcome of one trial does not affect the outcome of any other trial.
Examples
- Coin flipping: What is the probability of getting exactly 7 heads in 10 fair coin flips? (n = 10, p = 0.5, k = 7)
- Quality control: If 2% of products are defective, what is the probability that a batch of 50 contains exactly 3 defective items? (n = 50, p = 0.02, k = 3)
- Medical trials: If a treatment has a 70% success rate, what is the probability that 8 out of 10 patients respond positively? (n = 10, p = 0.7, k = 8)
- Surveys: If 60% of voters favor a candidate, what is the probability that exactly 15 out of 20 randomly selected voters support them? (n = 20, p = 0.6, k = 15)
Key Properties of the Binomial Distribution
Mean (Expected Value)
μ = n · p
The average number of successes you would expect over many repetitions of the experiment.
Variance
σ² = n · p · (1 − p)
Measures how spread out the distribution is around the mean. Maximized when p = 0.5.
Standard Deviation
σ = √(n · p · (1 − p))
The square root of the variance, giving a measure of spread in the same units as the data.
Skewness
(1 − 2p) / σ
When p < 0.5 the distribution is right-skewed; when p > 0.5 it is left-skewed; when p = 0.5 it is symmetric.
References
The formulas and concepts used in this calculator are based on established statistical theory:
Related Calculators
Note: This calculator uses exact combinatorial formulas for small values of n and logarithmic computation for larger values to maintain numerical accuracy. Results are rounded to display precision but computed at full floating-point accuracy. Always verify critical calculations with professional statistical software.
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