Binomial Distribution Calculator
Calculate the probability of k successes in n independent Bernoulli trials with probability p.
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Table of Contents
Comprehensive Guide to Binomial Distribution
What is Binomial Distribution?
The binomial distribution is one of the most fundamental and widely used probability distributions in statistics. It models the number of successes in a fixed number of independent experiments, each with the same probability of success.
Key Characteristics and Conditions
For a random experiment to follow a binomial distribution, it must satisfy these criteria:
- Fixed number of trials: The experiment consists of a fixed number (n) of trials.
- Independence: Each trial is independent of the others.
- Two outcomes: Each trial has exactly two possible outcomes ("success" or "failure").
- Constant probability: The probability of success (p) remains the same for each trial.
Applications of Binomial Distribution
Binomial distribution is applicable in numerous fields and scenarios:
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Quality Control: Testing whether products meet specifications.
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Medicine: Success rates of medical treatments or procedures.
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Finance: Probability of stock price movements or investment outcomes.
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Sports: Analyzing wins/losses in a series of games.
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Polling: Estimating the proportion of voters who favor a candidate.
Statistical Properties
Mean (Expected Value)
μ = n × p
Where n is the number of trials and p is the probability of success in each trial.
Variance
σ² = n × p × (1-p)
This measures the dispersion or spread of the distribution.
Standard Deviation
σ = √(n × p × (1-p))
The square root of the variance gives the standard deviation.
Skewness
(1-2p)/√(n×p×(1-p))
The distribution is symmetric when p=0.5, positively skewed when p<0.5, and negatively skewed when p>0.5.
Types of Binomial Probabilities
When working with binomial distributions, you can calculate several types of probabilities:
| Probability Type | Notation | Description |
|---|---|---|
| Exact | P(X = k) | Probability of exactly k successes |
| Cumulative (at most) | P(X ≤ k) | Probability of k or fewer successes |
| Cumulative (at least) | P(X ≥ k) | Probability of k or more successes |
| Range | P(a ≤ X ≤ b) | Probability of between a and b successes (inclusive) |
Relationship to Other Distributions
The binomial distribution connects to several other important distributions in statistics:
- Normal Approximation: For large n, the binomial distribution can be approximated by a normal distribution with mean μ=np and variance σ²=np(1-p).
- Bernoulli Distribution: A binomial distribution with n=1 is a Bernoulli distribution.
- Poisson Approximation: When n is large and p is small, the binomial distribution can be approximated by a Poisson distribution with parameter λ=np.
When to Use the Binomial Calculator
Use this binomial distribution calculator when you need to compute probabilities for situations involving:
- A fixed number of trials
- Independent events (the outcome of one trial doesn't affect others)
- Constant probability of success across all trials
- Only two possible outcomes per trial (success/failure)
Binomial Distribution Formula
The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success.
Where:
- P(X = k) is the probability of k successes
- C(n,k) is the number of combinations
- p is the probability of success
- n is the number of trials
- k is the number of successes
How to Calculate Binomial Probability
To calculate binomial probability, follow these steps:
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1Determine the number of trials (n)
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2Identify the number of successes (k)
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3Specify the probability of success (p)
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4Apply the binomial probability formula
Interpreting Binomial Probability
Understanding what the binomial probability tells you:
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1High Probability:
Indicates that the observed number of successes is likely to occur.
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2Low Probability:
Indicates that the observed number of successes is unlikely to occur.
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3Expected Value:
The expected number of successes is n * p.
Practical Examples
Example 1 Coin Toss
What is the probability of getting exactly 3 heads in 5 coin tosses?
n = 5, k = 3, p = 0.5
Probability = 0.3125
This means there's a 31.25% chance of getting exactly 3 heads.
Example 2 Test Questions
What is the probability of getting exactly 4 correct answers in a 10-question multiple-choice test (5 options per question)?
n = 10, k = 4, p = 0.2
Probability = 0.0881
This means there's an 8.81% chance of getting exactly 4 correct answers.
Example 3 Quality Control
What is the probability of finding exactly 2 defective items in a sample of 20 items, if the defect rate is 5%?
n = 20, k = 2, p = 0.05
Probability = 0.1887
This means there's an 18.87% chance of finding exactly 2 defective items.