Binomial Distribution Calculator
Calculate the probability of k successes in n independent Bernoulli trials with probability p.
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Table of Contents
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.