Probability Calculator
Probability Calculator
Calculate probabilities for single events, multiple events, combinations, permutations, and conditional probability with step-by-step solutions.
Probability Result
P = 0.5000
Probability Rule Applied:
Basic probability formula
Step-by-Step Calculation:
Probability Interpretation:
Moderate likelihood
Probability Visualization:
Probability quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).
What is Probability?
Probability is a mathematical measure of the likelihood that an event will occur. It's expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probability theory forms the foundation of statistics, risk assessment, and decision-making under uncertainty.
The basic probability formula is: P(A) = number of favorable outcomes / total number of possible outcomes
Probability Rules & Formulas
Basic Probability
f: favorable outcomes
n: total outcomes
Complement Rule
Probability event doesn't occur
P(not A)
Addition Rule
Probability of A OR B
Union of events
Multiplication Rule
Probability of A AND B
Intersection of events
Key Probability Concepts
| Concept | Notation | Formula | Example |
|---|---|---|---|
| Single Event | P(A) | favorable/total | P(rolling 3) = 1/6 |
| Complement | P(A') or P(Ā) | 1 - P(A) | P(not 3) = 5/6 |
| Union (OR) | P(A∪B) | P(A)+P(B)-P(A∩B) | P(3 or even) = 4/6 |
| Intersection (AND) | P(A∩B) | P(A)×P(B) if independent | P(3 and heads) = 1/12 |
| Conditional | P(A|B) | P(A∩B)/P(B) | P(3|odd) = 1/3 |
| Independent Events | A ⊥ B | P(A∩B)=P(A)P(B) | Coin flips |
Step-by-Step Examples
Example 1: Single Die Roll
- Event: Rolling a 3 on a fair six-sided die
- Favorable outcomes: 1 (only the number 3)
- Total outcomes: 6 (numbers 1 through 6)
- Probability: P(3) = 1/6 ≈ 0.1667
- Percentage: 16.67%
- Odds: 1:5 (1 for, 5 against)
Example 2: Drawing Cards
- Event: Drawing an Ace from a standard 52-card deck
- Favorable outcomes: 4 (Ace of each suit)
- Total outcomes: 52 (total cards)
- Probability: P(Ace) = 4/52 = 1/13 ≈ 0.0769
- Percentage: 7.69%
- Odds: 1:12 (1 for, 12 against)
Example 3: Coin Toss (Multiple Events)
- Event: Getting heads on two consecutive coin tosses
- P(heads on first toss) = 1/2
- P(heads on second toss) = 1/2
- Since independent: P(H and H) = (1/2) × (1/2) = 1/4
- Probability: 0.25 or 25%
- Odds: 1:3
Common Probability Scenarios
Gambling & Games
- Dice games: Craps probabilities, Yahtzee combinations
- Card games: Poker hand probabilities, blackjack odds
- Lotteries: Winning number probabilities, expected value
- Roulette: Red/black, odd/even probabilities
Real-World Applications
- Risk assessment: Insurance premiums, disaster probabilities
- Quality control: Defect rates in manufacturing
- Medical testing: Disease prevalence, test accuracy
- Weather forecasting: Precipitation probabilities
- Sports analytics: Win probabilities, player performance
Everyday Decisions
- Commute times: Probability of traffic delays
- Product reliability: Chance of device failure
- Financial planning: Investment risk assessment
- Health decisions: Treatment success probabilities
Combinations vs Permutations
| Aspect | Combinations (nCr) | Permutations (nPr) |
|---|---|---|
| Order matters? | No | Yes |
| Formula | n! / [r!(n-r)!] | n! / (n-r)! |
| Notation | C(n,r) or ⁿCᵣ | P(n,r) or ⁿPᵣ |
| Example | Choosing committee members | Arranging books on shelf |
| When to use | Selection without order | Arrangement with order |
Conditional Probability & Bayes' Theorem
Conditional probability P(A|B) is the probability of event A occurring given that event B has occurred. The formula is:
Bayes' Theorem relates conditional probabilities:
This is fundamental in medical testing, machine learning, and statistical inference.
Related Calculators
Frequently Asked Questions (FAQs)
Q: What's the difference between probability and odds?
A: Probability is the ratio of favorable outcomes to total outcomes (P = f/n). Odds are the ratio of favorable to unfavorable outcomes (odds = f:(n-f)). For example, probability 1/4 = 0.25 corresponds to odds 1:3.
Q: How do I calculate probability for dependent events?
A: For dependent events, use P(A∩B) = P(A) × P(B|A). First calculate P(A), then P(B given A has occurred). This accounts for the dependency between events.
Q: What does P(A|B) mean in conditional probability?
A: P(A|B) means "the probability of A given B." It's the probability that event A occurs, given that we know event B has already occurred. This updates our probability based on new information.
Q: When should I use combinations vs permutations?
A: Use combinations when order doesn't matter (choosing committee members, lottery numbers). Use permutations when order matters (passwords, race rankings, seating arrangements).
Master probability calculations with Toolivaa's free Probability Calculator, and explore more mathematical tools in our Math Calculators collection.