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MonoCalc

Probability Calculator

Number of Favorable Outcomes
The number of outcomes you're interested in
Total Number of Outcomes
The total number of possible outcomes

Result

Enter values to see results

About the tool

🎲 Probability Calculator – Calculate Simple & Compound Probability

Need to calculate probability? The Probability Calculator helps you compute simple probability, complementary probability, and complex compound probability scenarios with step-by-step solutions.

This guide explains what probability is, how to calculate it, and walks you through using our free online probability calculator for various probability problems including independent events, dependent events, and conditional probability.

📘 What is Probability?

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible, 1 means the event is certain, and values in between represent varying degrees of likelihood.

Probability can also be expressed as a percentage (0% to 100%) or as a fraction. For example, the probability of flipping heads on a fair coin is 0.5, 50%, or 1/2.

⚙️ How the Probability Calculator Works

Our probability calculator offers three main calculation modes:

  1. Simple Probability - Calculate basic probability from favorable and total outcomes
  2. Complementary Probability - Calculate the probability that an event does NOT occur
  3. Compound Probability - Calculate probability for multiple events (AND, OR, conditional)

🧩 Key Features

  • ⚡ Instant calculations for all probability types
  • 📊 Step-by-step solutions with formulas
  • 🔢 Multiple format support (decimal, fraction, percentage)
  • 🎯 Independent, dependent, and conditional event calculations
  • 🎲 Random example generator for practice
  • 📱 Export results as text files
  • 🔐 Client-side only — no data is ever uploaded

💡 Probability Formulas

1. Simple Probability

The basic probability formula is:

P(A) = Number of Favorable Outcomes / Total Number of Possible Outcomes

Example: What is the probability of rolling a 3 on a standard die?
Favorable outcomes: 1 (only one face shows 3)
Total outcomes: 6 (die has 6 faces)
P(rolling 3) = 1/6 ≈ 0.1667 or 16.67%

2. Complementary Probability

The complementary probability is the probability that an event does NOT occur:

P(A') = 1 - P(A)

Example: If the probability of rain is 0.3 (30%), what is the probability of no rain?
P(no rain) = 1 - 0.3 = 0.7 or 70%

3. Compound Probability - Independent Events (AND)

For independent events (one event doesn't affect the other):

P(A and B) = P(A) × P(B)

Example: What is the probability of flipping heads twice in a row?
P(heads) = 0.5
P(heads and heads) = 0.5 × 0.5 = 0.25 or 25%

4. Compound Probability - Mutually Exclusive Events (OR)

For mutually exclusive events (events that cannot occur simultaneously):

P(A or B) = P(A) + P(B)

Example: What is the probability of rolling a 2 or a 5 on a die?
P(2) = 1/6, P(5) = 1/6
P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3 ≈ 0.333 or 33.33%

5. Compound Probability - Non-Mutually Exclusive Events (OR)

For events that can occur together:

P(A or B) = P(A) + P(B) - P(A and B)

Example: Drawing a card that is either red or a face card from a standard deck:
P(red) = 26/52, P(face card) = 12/52, P(red face card) = 6/52
P(red or face card) = 26/52 + 12/52 - 6/52 = 32/52 ≈ 0.615 or 61.5%

6. Conditional Probability

The probability of event A occurring given that event B has occurred:

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

Example: What is the probability of drawing a king given that you've drawn a face card?
P(king and face card) = 4/52
P(face card) = 12/52
P(king|face card) = (4/52) / (12/52) = 4/12 = 1/3 ≈ 0.333 or 33.33%

7. Dependent Events (AND)

For dependent events (where one event affects the probability of another):

P(A and B) = P(A) × P(B|A)

Example: Drawing two aces from a deck without replacement:
P(first ace) = 4/52
P(second ace | first ace) = 3/51
P(two aces) = (4/52) × (3/51) ≈ 0.0045 or 0.45%

🌟 Practical Applications of Probability

  • 🎰 Gaming & Gambling: Calculating odds in dice games, card games, lottery, and casino games
  • 📊 Statistics: Hypothesis testing, confidence intervals, and data analysis
  • 🏥 Medicine: Disease risk assessment, drug efficacy studies, diagnostic testing
  • 💼 Business: Risk management, quality control, market forecasting
  • 🌦️ Weather: Precipitation forecasts, climate modeling
  • 🔬 Science: Quantum mechanics, genetics, experimental design
  • 💻 Machine Learning: Bayesian networks, probabilistic models, classification algorithms

🔄 How to Use the Probability Calculator

Simple Probability Mode:

  1. Enter the number of favorable outcomes (outcomes you're interested in)
  2. Enter the total number of possible outcomes
  3. View the calculated probability in decimal, fraction, and percentage formats
  4. Review the step-by-step solution
  5. Use "Random Example" to generate practice problems

Complementary Probability Mode:

  1. Enter the probability of the event occurring (P(A))
  2. The calculator will compute P(A') = 1 - P(A)
  3. View the result and step-by-step calculation

Compound Probability Mode:

  1. Select the type of compound event (independent AND, mutually exclusive OR, etc.)
  2. Enter the required probability values based on the event type
  3. View the calculated compound probability
  4. Review the formula and step-by-step solution
  5. Export results as a text file if needed

✅ Understanding Probability Concepts

Independent vs Dependent Events

Independent events are events where the occurrence of one does not affect the probability of the other. For example, flipping a coin twice - the second flip is not affected by the first.

Dependent events are events where the occurrence of one affects the probability of the other. For example, drawing cards without replacement - the second draw's probability changes based on what was drawn first.

Mutually Exclusive vs Non-Mutually Exclusive Events

Mutually exclusive events cannot occur at the same time. For example, rolling a 2 and rolling a 5 on a single die roll.

Non-mutually exclusive events can occur simultaneously. For example, drawing a red card and drawing a face card from a deck (red face cards exist).

🎯 Tips for Working with Probability

  • Always ensure probabilities are between 0 and 1 (or 0% to 100%)
  • For compound probability, identify whether events are independent or dependent
  • When using the OR rule, check if events are mutually exclusive
  • Verify that all possible outcomes have been counted
  • Use complementary probability for "at least one" problems: P(at least one) = 1 - P(none)
  • Draw probability trees for complex multi-step problems
  • Remember that probability does not guarantee outcomes in small sample sizes

📚 Common Probability Examples

Coin Flips

Probability of heads: 1/2 or 0.5 or 50%
Probability of heads three times in a row: (1/2)³ = 1/8 = 0.125 or 12.5%

Dice Rolls

Probability of rolling a specific number: 1/6 ≈ 0.167 or 16.67%
Probability of rolling an even number: 3/6 = 1/2 = 0.5 or 50%
Probability of rolling a number greater than 4: 2/6 = 1/3 ≈ 0.333 or 33.33%

Card Drawing

Probability of drawing an ace: 4/52 = 1/13 ≈ 0.077 or 7.7%
Probability of drawing a heart: 13/52 = 1/4 = 0.25 or 25%
Probability of drawing a red card: 26/52 = 1/2 = 0.5 or 50%

🔍 Advanced Probability Concepts

For more complex probability scenarios, you may need to apply:

  • Bayes' Theorem: For updating probabilities based on new information
  • Permutations and Combinations: For calculating total possible outcomes
  • Probability Distributions: Binomial, Poisson, Normal distributions for repeated trials
  • Law of Large Numbers: Understanding that probability becomes more accurate with larger samples

🚀 Why Use Our Probability Calculator?

  • ✅ Supports all major probability calculation types
  • ✅ Shows detailed step-by-step solutions for learning
  • ✅ Multiple output formats (decimal, fraction, percentage)
  • ✅ No registration or installation required
  • ✅ Works on all devices (desktop, tablet, mobile)
  • ✅ Completely free to use
  • ✅ Export and save your results

Whether you're a student learning probability theory, a professional analyzing risk scenarios, or just curious about the chances of an event occurring, our Probability Calculator provides accurate calculations with clear explanations to help you understand the mathematics behind probability.

Frequently Asked Questions

  • Is the Probability Calculator free ?

    Yes, Probability Calculator is totally free :)

  • Can i use the Probability Calculator offline ?

    Yes, you can install the webapp as PWA.

  • Is it safe to use Probability Calculator ?

    Yes, any data related to Probability Calculator only stored in your browser(if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

  • What is probability and how is it calculated?

    Probability is a measure of the likelihood that an event will occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. The result is expressed as a number between 0 (impossible) and 1 (certain), or as a percentage between 0% and 100%.

  • What is the difference between independent and dependent events?

    Independent events are events where the occurrence of one does not affect the probability of the other (e.g., flipping two coins). Dependent events are events where the occurrence of one affects the probability of the other (e.g., drawing cards from a deck without replacement). For independent events, P(A and B) = P(A) × P(B), while for dependent events, you must account for the changed sample space.

  • How do you calculate compound probability?

    Compound probability involves calculating the probability of two or more events occurring. For 'AND' (intersection): multiply the probabilities if independent, or adjust for dependence. For 'OR' (union): add the probabilities and subtract the probability of both occurring together: P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events, P(A or B) = P(A) + P(B).

  • What is complementary probability?

    Complementary probability is the probability that an event does NOT occur. It is calculated as 1 minus the probability that the event does occur: P(A') = 1 - P(A). For example, if the probability of rain is 0.3 (30%), the probability of no rain is 0.7 (70%).

  • What is conditional probability and how is it calculated?

    Conditional probability is the probability of an event A occurring given that another event B has already occurred. It is denoted as P(A|B) and calculated as P(A|B) = P(A and B) / P(B), provided P(B) > 0. This is useful for calculating probabilities in dependent scenarios.

  • Can probability be greater than 1 or less than 0?

    No, probability must always be between 0 and 1 (inclusive). A probability of 0 means the event is impossible, a probability of 1 means the event is certain, and any value in between represents the likelihood. If your calculation results in a value outside this range, there is an error in your inputs or calculation.