Probability Distribution Calculator

Calculate probabilities for all major distributions with visualizations and step-by-step solutions

Distribution

Calculation Type

Number of Trials (n)

Please enter a positive integer

Probability of Success (p)

Please enter a value between 0 and 1

Number of Successes (k)

Please enter a non-negative integer

Mean (μ)

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Standard Deviation (σ)

Please enter a positive number

Value (x)

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Upper Value

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Rate (λ)

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Number of Events (k)

Please enter a non-negative integer

Rate (λ)

Please enter a positive number

Value (x)

Please enter a non-negative number

Lower Bound (a)

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Upper Bound (b)

Please enter a number greater than lower bound

Value (x)

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Probability of Success (p)

Please enter a value between 0 and 1

Trial Number (k)

Please enter a positive integer

Population Size (N)

Please enter a positive integer

Success States (K)

Please enter a positive integer

Sample Size (n)

Please enter a positive integer

Successes in Sample (k)

Please enter a non-negative integer

Number of Successes (r)

Please enter a positive integer

Probability of Success (p)

Please enter a value between 0 and 1

Number of Failures (k)

Please enter a non-negative integer

Probability Result

Select distribution and enter parameters

Step-by-Step Solution

Step 1:

Identify the distribution and parameters

Recognize the distribution type and input values

Step 2:

Apply the appropriate probability formula

Use the formula for the selected distribution

Step 3:

Calculate the probability

Perform the necessary calculations

Step 4:

Interpret the result

Explain what the probability means

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Formula: $P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}$

Parameters:

n = number of trials
p = probability of success on each trial
k = number of successes

Applications: Quality control, survey analysis, medical trials, game theory

Probability Distributions Explained

Probability distributions describe how probabilities are distributed over the values of a random variable. Here's a comprehensive overview:

Distribution	Type	Parameters	Probability Mass/Density Function	Applications
Binomial	Discrete	n, p	$P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}$	Success/failure experiments, quality control
Normal	Continuous	μ, σ	$f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}}$	Natural phenomena, measurement errors, IQ scores
Poisson	Discrete	λ	$P (X = k) = \frac{λ^{k} e^{- λ}}{k!}$	Rare events, arrivals in queues, radioactive decay
Exponential	Continuous	λ	$f (x) = λ e^{- λ x}$	Time between events, survival analysis
Uniform	Continuous	a, b	$f (x) = \frac{1}{b - a}$	Random number generation, simulations
Geometric	Discrete	p	$P (X = k) = (1 - p)^{k - 1} p$	Number of trials until first success

Key Concepts:

Probability Mass Function (PMF): For discrete distributions, gives probability at each point
Probability Density Function (PDF): For continuous distributions, probability is area under curve
Cumulative Distribution Function (CDF): Probability that X ≤ x
Mean (μ): Expected value of the distribution
Variance (σ²): Measure of spread around the mean
Standard Deviation (σ): Square root of variance

Applications

Quality Control: Binomial distribution for defect rates
Finance: Normal distribution for stock returns
Telecommunications: Poisson for call arrivals
Reliability Engineering: Exponential for failure times
Manufacturing: Uniform for tolerances
Clinical Trials: Geometric for trial success
Quality Testing: Hypergeometric for sampling
Epidemiology: Negative binomial for disease spread

Frequently Asked Questions

Common questions about probability distributions:

Discrete Distributions:

Random variable takes countable values
Probability defined at specific points
Examples: Binomial, Poisson, Geometric
Probability Mass Function (PMF)

Continuous Distributions:

Random variable takes uncountable values
Probability defined over intervals
Examples: Normal, Exponential, Uniform
Probability Density Function (PDF)

Binomial Distribution:

Fixed number of trials (n)
Constant probability of success (p)
Trials are independent
Examples: Coin flips, pass/fail tests

Poisson Distribution:

Events occur at a constant rate (λ)
Events are independent
Number of events in fixed interval
Examples: Calls to call center, website visits

Rule of thumb: Poisson approximates binomial when n is large (n > 50) and p is small (p < 0.1)

The normal distribution (Gaussian distribution) is symmetric and bell-shaped:

68% of values within 1 standard deviation of mean
95% within 2 standard deviations
99.7% within 3 standard deviations

Key properties:

Mean = Median = Mode
Completely determined by mean (μ) and standard deviation (σ)
Many statistical tests assume normality
Central Limit Theorem: Sums of random variables tend to normal

The exponential and Poisson distributions are closely related:

Poisson distribution: Models number of events in fixed time period
Exponential distribution: Models time between events
If events follow Poisson process with rate λ, then time between events follows exponential distribution with parameter λ
Both distributions are memoryless

Example: If customers arrive at a store following Poisson distribution with rate 10 per hour, then time between arrivals follows exponential distribution with mean 6 minutes.

Probability Rules

Addition Rule: P(A∪B) = P(A) + P(B) - P(A∩B)
Multiplication Rule: P(A∩B) = P(A) * P(B|A)
Complement Rule: P(A') = 1 - P(A)
Conditional Probability: P(A|B) = P(A∩B)/P(B)
Bayes' Theorem: P(A|B) = [P(B|A) * P(A)] / P(B)
Total Probability: P(B) = Σ P(B|A_i) * P(A_i)

Statistical Measures

Mean: μ = Σx_i / n
Variance: σ² = Σ(x_i - μ)² / n
Standard Deviation: σ = √σ²
Covariance: Cov(X,Y) = Σ(x_i - μ_x)(y_i - μ_y) / n
Correlation: ρ = Cov(X,Y) / (σ_x σ_y)
Skewness: measure of asymmetry
Kurtosis: measure of tail heaviness

Probability Distribution Calculator

Binomial Distribution

Probability Distributions Explained

Applications

Frequently Asked Questions

What is the difference between discrete and continuous distributions?

When should I use binomial vs Poisson distribution?

How do I interpret the normal distribution?

What is the relationship between exponential and Poisson distributions?

Probability Rules

Statistical Measures

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