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Enter the number of trials and successes, probability, and select the condition to calculate the event's probability, standard deviation, variance, and mean, with detailed calculations and graphical interpretation displayed.
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An online Binomial Distribution Calculator helps you compute exact and cumulative probabilities for a given number of trials. It can also determine the mean, variance, and standard deviation of a binomial distribution instantly. Below, you’ll learn what the binomial distribution is, when to use it, and how to calculate probabilities step-by-step.
The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes:
Each trial must have the same probability of success (p), and the trials must be independent.
The distribution depends on two parameters:
For example, if a basketball player attempts 8 free throws and the probability of scoring each shot is 0.7, then:
n = 8, p = 0.7
The probability of getting exactly r successes in n trials is:
$$ P(X = r) = \binom{n}{r} p^r (1 - p)^{n-r} $$
Where:
Problem:
A machine produces items with a 20% defect rate. If 6 items are selected, what is the probability that exactly 2 are defective?
Given:
Step 1: Apply formula
$$ P(2) = \binom{6}{2} (0.20)^2 (0.80)^4 $$
Step 2: Compute combination
$$ \binom{6}{2} = \frac{6!}{2!4!} = 15 $$
Step 3: Substitute values
$$ P(2) = 15 × (0.04) × (0.4096) $$
$$ P(2) = 15 × 0.016384 $$
$$ P(2) ≈ 0.24576 $$
Answer: The probability of exactly 2 defective items is approximately 0.246.
For a binomial distribution:
$$ \text{Mean } (\mu) = np $$
$$ \text{Variance } (\sigma^2) = np(1-p) $$
$$ \text{Standard Deviation } (\sigma) = \sqrt{np(1-p)} $$
Using the above example:
A binomial experiment must satisfy these conditions:
A single success/failure experiment is called a Bernoulli trial. Multiple Bernoulli trials form a binomial distribution.
The negative binomial distribution measures the number of trials needed to achieve a fixed number of successes. It differs from the binomial distribution, which counts successes in a fixed number of trials.
When n is large and p is not extremely small or large, the binomial distribution approximates the normal distribution.
The binomial distribution is a powerful statistical tool for analyzing experiments with two possible outcomes. By entering the values of n, p, and r, a binomial distribution calculator quickly determines probabilities, mean, variance, and standard deviation—making complex probability calculations simple and accurate.
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