Binomial Distribution Calculator
| k | P(X = k) | P(X <= k) |
|---|
Understanding Binomial Distribution
What is a Binomial Distribution?
A binomial distribution models the probability of obtaining a fixed 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 commonly used discrete probability distributions.
The Binomial Formula
The probability of getting exactly k successes in n trials is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
C(n, k) = n! / (k! * (n-k)!)is the binomial coefficientnis the number of trialskis the number of successespis the probability of success on each trial
When to Use Binomial Distribution
The binomial distribution is appropriate when:
- There is a fixed number of trials (n)
- Each trial is independent
- Each trial has only two possible outcomes (success or failure)
- The probability of success (p) is constant for all trials
Common Applications
- Quality Control: Number of defective items in a batch
- Medical Testing: Number of patients responding to treatment
- Survey Analysis: Number of people answering "yes" to a question
- Games of Chance: Number of heads in coin flips
- Marketing: Number of customers making a purchase
Properties and Statistics
- Mean:
mu = n * p - Variance:
sigma^2 = n * p * (1-p) - Standard Deviation:
sigma = sqrt(n * p * (1-p))
Normal Approximation
When n is large and p is not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution with mean np and standard deviation sqrt(np(1-p)). The rule of thumb is to use normal approximation when both np >= 5 and n(1-p) >= 5.
Examples
Example 1: A fair coin is flipped 10 times. What is the probability of getting exactly 6 heads?
Here, n=10, p=0.5, k=6. Using the formula: P(X=6) = C(10,6) * 0.5^6 * 0.5^4 = 210 * 0.015625 * 0.0625 = 0.2051 or about 20.51%
Example 2: A company produces items with a 2% defect rate. In a sample of 100 items, what is the probability of finding at most 3 defective items?
Here, n=100, p=0.02. We need P(X<=3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.8590 or about 85.90%
Frequently Asked Questions
What is a binomial distribution?
A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. For example, flipping a coin 10 times and counting heads.
What is the formula for binomial probability?
The binomial probability formula is P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where C(n,k) is the binomial coefficient n!/(k!(n-k)!), n is the number of trials, k is the number of successes, and p is the probability of success.
When should I use a binomial distribution?
Use a binomial distribution when you have a fixed number of independent trials, each trial has only two outcomes (success or failure), and the probability of success is constant across all trials.
What is the difference between P(X=k) and P(X<=k)?
P(X=k) is the exact probability of getting exactly k successes, while P(X<=k) is the cumulative probability of getting k or fewer successes. P(X<=k) is the sum of all probabilities from P(X=0) to P(X=k).
What is the mean of a binomial distribution?
The mean (expected value) of a binomial distribution is n*p, where n is the number of trials and p is the probability of success. For example, if you flip a coin 10 times, the expected number of heads is 10*0.5 = 5.
When can I approximate a binomial distribution with a normal distribution?
A binomial distribution can be approximated by a normal distribution when both np >= 5 and n(1-p) >= 5. This is useful for large values of n where exact calculations become computationally intensive.
What is the variance of a binomial distribution?
The variance of a binomial distribution is n*p*(1-p), where n is the number of trials and p is the probability of success. The standard deviation is the square root of the variance.
How do I calculate binomial coefficients?
The binomial coefficient C(n,k) = n! / (k! * (n-k)!) represents the number of ways to choose k items from n items. For example, C(10,3) = 10! / (3! * 7!) = 120.
Related Tools
Privacy & Limitations
- All calculations run entirely in your browser -- nothing is sent to any server.
- Results are computed using standard formulas and should be verified for critical applications.
Related Tools
View all toolsPercentage Calculator
Calculate percentages, find X% of Y, percentage change
Ratio Calculator
Simplify a ratio and see the reduced form
Number to Words Converter
Convert numbers to written English words
Scientific Notation Converter
Convert between standard and scientific notation
Decimal to Fraction Converter
Convert decimals to fractions and vice versa
Quadratic Formula Calculator
Solve ax² + bx + c = 0 with step-by-step solution
Binomial Distribution Calculator FAQ
What is a binomial distribution?
A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. For example, flipping a coin 10 times and counting heads.
What is the formula for binomial probability?
The binomial probability formula is P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where C(n,k) is the binomial coefficient n!/(k!(n-k)!), n is the number of trials, k is the number of successes, and p is the probability of success.
When should I use a binomial distribution?
Use a binomial distribution when you have a fixed number of independent trials, each trial has only two outcomes (success or failure), and the probability of success is constant across all trials.
What is the difference between P(X=k) and P(X<=k)?
P(X=k) is the exact probability of getting exactly k successes, while P(X<=k) is the cumulative probability of getting k or fewer successes. P(X<=k) is the sum of all probabilities from P(X=0) to P(X=k).
What is the mean of a binomial distribution?
The mean (expected value) of a binomial distribution is n*p, where n is the number of trials and p is the probability of success. For example, if you flip a coin 10 times, the expected number of heads is 10*0.5 = 5.
When can I approximate a binomial distribution with a normal distribution?
A binomial distribution can be approximated by a normal distribution when both np >= 5 and n(1-p) >= 5. This is useful for large values of n where exact calculations become computationally intensive.
What is the variance of a binomial distribution?
The variance of a binomial distribution is n*p*(1-p), where n is the number of trials and p is the probability of success. The standard deviation is the square root of the variance.