Probability Calculator – Basic, Binomial, Combinations, Permutations, and Normal Distribution

  • Adam
  • November 4, 2025

Free online probability calculator for basic probability, permutations, combinations, conditional probability, binomial distribution, and normal distribution with step-by-step solutions.

Probability Calculator

Advanced Tool for Computing Probabilities, Distributions, and Statistical Events

Quick Navigation

Basic Probability

P(A) = Favorable Outcomes / Total Possible Outcomes

Combinations (Binomial Coefficient)

C(n,k) = n! / (k! × (n-k)!)

Permutations

P(n,k) = n! / (n-k)!

Conditional Probability

P(A|B) = P(A and B) / P(B)

Binomial Probability

P(X=k) = C(n,k) × p^k × (1-p)^(n-k)

Normal Distribution

Calculate probability P(X ≤ x) for normal distribution

What is Probability?

Probability is the mathematical study of likelihood and chance. It quantifies how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain), or as a percentage between 0% and 100%. Probability is fundamental to statistics, science, finance, and decision-making under uncertainty.

There are three main interpretations: Classical (mathematical theory), Empirical (based on observations), and Subjective (personal belief). Different probability concepts include independent events (one doesn't affect another), dependent events (outcomes are related), and mutually exclusive events (can't both happen). Understanding these distinctions is crucial for correct probability calculations.

This calculator helps you compute probabilities for various scenarios: basic probability, permutations and combinations for counting arrangements, conditional probability for events that depend on each other, binomial probability for repeated yes/no trials, and normal distribution for continuous data. Each type has different applications in real-world problems.

Key Concept: Probability represents uncertainty. A probability of 0.5 (50%) doesn't mean exactly 50% will occur in a small sample—it means outcomes approach 50% as sample size increases (Law of Large Numbers).

Key Features & Capabilities

This comprehensive probability calculator provides complete analysis for different probability types:

📊 Basic Probability Simple favorable/total outcomes calculation
🎲 Combinations Binomial coefficients C(n,k)
🔄 Permutations Ordered arrangements P(n,k)
🔗 Conditional Probability P(A|B) dependent events
🎯 Binomial Distribution Exactly k successes in n trials
📈 Normal Distribution Continuous probability calculations
📋 Multiple Formats Decimal, percentage, odds
📋 Step-by-Step Detailed calculation breakdown
🧮 Z-Score Conversion Normal distribution standardization
📋 Copy Function One-click copy to clipboard
🎓 Educational Learn probability concepts
📱 Responsive Works on all devices

How to Use This Calculator

Step-by-Step Guide

  1. Identify Problem Type: Determine which calculator tab you need: Basic, Combinations, Permutations, Conditional, Binomial, or Normal Distribution.
  2. Gather Information: Collect all necessary data: number of outcomes, total possibilities, probabilities, number of trials, etc.
  3. Select Tab: Click on the appropriate calculator tab for your problem.
  4. Enter Values: Input all required parameters in the form fields.
  5. Click Calculate: Press Calculate to compute the probability.
  6. Review Results: See result in decimal, percentage, and detailed calculation steps.
  7. Understand Calculation: Study step-by-step breakdown to understand the mathematics.
  8. Copy or Continue: Use Copy for results or Clear to calculate new problem.

Tips for Accurate Use

  • Input Format: Percentages should be 0-100, not decimals 0-1. Calculator automatically converts.
  • Order Matters: Use Permutations for arrangements where order matters (arrangements, sequences). Use Combinations when order doesn't matter (selections, subsets).
  • Binomial Requirements: Probability p must be between 0-100%. k cannot exceed n. Use for yes/no repeated trials.
  • Normal Distribution: Works for any mean and standard deviation. Automatically calculates z-score and uses standard normal table.
  • Large Numbers: Factorial calculations work efficiently up to n=100. Results shown in scientific notation if needed.

Complete Formulas Guide

Basic Probability

Simple Probability Formula
P(A) = Favorable Outcomes / Total Possible Outcomes

Example: Drawing heart from standard deck
Favorable = 13 hearts
Total = 52 cards
P(heart) = 13/52 = 0.25 or 25%

Combinations

Binomial Coefficient Formula
C(n,k) = n! / (k! × (n-k)!)

Example: Ways to choose 3 from 10
C(10,3) = 10! / (3! × 7!)
= 3,628,800 / (6 × 5,040)
= 120 combinations

Permutations

Permutation Formula
P(n,k) = n! / (n-k)!

Example: Arrange 3 from 10
P(10,3) = 10! / 7!
= 10 × 9 × 8
= 720 permutations

Conditional Probability

Conditional Probability Formula
P(A|B) = P(A and B) / P(B)

Probability of A given B occurred
Example: Probability of rain given clouds present
P(rain|clouds) = P(rain and clouds) / P(clouds)

Binomial Probability

Binomial Distribution Formula
P(X=k) = C(n,k) × p^k × (1-p)^(n-k)

Example: Exactly 5 heads in 10 coin flips
P(X=5) = C(10,5) × 0.5^5 × 0.5^5
= 252 × 0.03125 × 0.03125
≈ 0.246 or 24.6%

Normal Distribution

Z-Score and Normal CDF
Z = (X - μ) / σ

Convert value to standard normal distribution
Then look up Z in standard normal table
Example: X=110, mean=100, SD=15
Z = (110-100)/15 = 0.667

Understanding Probability Concepts

Independent vs. Dependent Events

Independent events are unrelated—one's outcome doesn't affect the other. Flipping a coin twice: first flip doesn't affect second. Dependent events are related: drawing without replacement. Picking card changes deck composition for next pick. Different formulas apply to each type.

Mutually Exclusive Events

Events that cannot both happen simultaneously. Drawing single card: can't be both heart and spade. For mutually exclusive events: P(A or B) = P(A) + P(B). Key distinction from inclusive events where both can occur.

Permutations vs. Combinations

Permutations consider order: arranging 3 people in line (1-2-3 differs from 2-1-3). Combinations ignore order: selecting 3 people from group (selection {1,2,3} same as {2,1,3}). Permutations always produce larger numbers since order creates more arrangements.

Binomial Distribution

Used for experiments with two outcomes (success/failure, yes/no, heads/tails) repeated independently. Requires: fixed number of trials, constant probability, independent trials. Example: quality control testing 100 items for defects.

Normal Distribution

Bell-shaped curve describing many natural phenomena (heights, weights, test scores). Defined by mean (center) and standard deviation (spread). About 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD of mean (empirical rule).

Worked Examples

Example 1: Basic Probability

Problem: What's probability of rolling a 6 on a fair die?

Solution:
Favorable outcomes: 1 (just one 6)
Total outcomes: 6 (faces 1-6)

P(6) = 1/6 ≈ 0.1667 or 16.67%

Example 2: Combinations

Problem: Lottery picking 6 numbers from 49. How many combinations possible?

Solution:
C(49,6) = 49! / (6! × 43!)
= 49 × 48 × 47 × 46 × 45 × 44 / (6 × 5 × 4 × 3 × 2 × 1)
= 10,068,347,520 / 720
≈ 13,983,816 combinations

Probability of winning = 1 in 13,983,816!

Example 3: Conditional Probability

Problem: Medical test: 95% accurate. Disease affects 1% of population. If test is positive, what's true probability you have disease?

Solution:
Uses Bayes' Theorem
P(disease|positive) = P(positive|disease) × P(disease) / P(positive)

Result: ≈ 16% (not 95%!)
Most positives are false positives because disease is rare

Example 4: Binomial Probability

Problem: Fair coin flipped 10 times. Probability of exactly 7 heads?

Solution:
n = 10 trials
k = 7 successes (heads)
p = 0.5 (fair coin)

P(X=7) = C(10,7) × 0.5^7 × 0.5^3
= 120 × 0.0078125 × 0.125
≈ 0.117 or 11.7%

Example 5: Normal Distribution

Problem: Heights normally distributed: mean = 70 inches, SD = 3. Probability height ≤ 73?

Solution:
Z = (73 - 70) / 3 = 1.0

Z-score of 1.0 corresponds to
cumulative probability ≈ 0.8413 or 84.13%

About 84% of people ≤ 73 inches tall

Frequently Asked Questions

❓ What's the difference between odds and probability?
Probability ranges 0-1 (or 0%-100%). Odds are ratio of favorable to unfavorable outcomes. Probability 0.25 = odds of 1:3 (one favorable per three unfavorable). Odds are often used in gambling and betting.
❓ Can probability be greater than 1?
No. Probability is always 0 ≤ P ≤ 1. 0 means impossible, 1 means certain. Values outside this range are invalid. Be careful with odds notation which can exceed 1.
❓ What's the gambler's fallacy?
Mistaken belief that past results affect future probability. Coin that landed heads 5 times has same 50% probability heads on flip 6. Each trial is independent; history doesn't influence future.
❓ Why does binomial distribution have n choose k?
C(n,k) counts arrangements of k successes among n trials. Multiply by p^k × (1-p)^(n-k) for probability of each arrangement. Different arrangements have same probability, so multiply by count.
❓ When is normal distribution appropriate?
Normal distribution works for continuous data: heights, weights, test scores, measurements. Central Limit Theorem: averages of samples approach normal regardless of original distribution.
❓ What does standard deviation represent?
Standard deviation measures spread of data. About 68% falls within 1 SD of mean in normal distribution. Larger SD = wider spread = more variation. Important for normal distribution calculations.
❓ How are probability and statistics related?
Probability is theoretical (what should happen). Statistics is empirical (what actually happens). Probability forms foundation for statistical inference and hypothesis testing.
❓ What's the law of large numbers?
As sample size increases, observed results approach theoretical probability. Coin flips approach 50% heads with more flips. Explains why large samples give more reliable estimates.
❓ Can I multiply probabilities?
Only for independent events. P(A and B) = P(A) × P(B). For dependent events, use conditional probability: P(A and B) = P(A) × P(B|A). Don't multiply unless events are independent!
❓ What's Bayes' Theorem?
Formula for updating probabilities given new evidence. P(A|B) = P(B|A) × P(A) / P(B). Essential for medical diagnosis, spam detection, and conditional probability problems.

Calculate Probabilities

Whether you're analyzing games of chance, assessing risk, predicting outcomes, or learning probability theory, this comprehensive probability calculator provides instant solutions with complete analysis. Fast, accurate, and completely free.

Related Calculators