Probability Calculator
Calculate probability for single events, multiple events (AND, OR), conditional probability, and at least one occurrence.
Calculate Probability
Calculation Type:
P(A) = 3 / 10
Probability
30.0000%
= 0.300000 = 3/10
Visual Representation:
Probability Formulas
Basic Probability
P(A) = favorable / total
Complement
P(not A) = 1 - P(A)
Addition Rule
P(A∪B) = P(A) + P(B) - P(A∩B)
At Least One
P(≥1) = 1 - (1-p)^n
What Is Probability?
Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1 (or 0% to 100%). It's the foundation of statistics, risk assessment, machine learning, and decision-making under uncertainty. Understanding probability helps interpret everything from weather forecasts to medical test results.
| Probability Value | Meaning | Example |
|---|---|---|
| P = 0 | Impossible event | Rolling 7 on a standard die |
| P = 0.5 (50%) | Equally likely outcomes | Fair coin showing heads |
| P = 1 | Certain event | Rolling 1-6 on a standard die |
| 0 < P < 0.5 | Unlikely but possible | Rolling a 6 (P = 1/6 ≈ 0.17) |
| 0.5 < P < 1 | Likely but not certain | Not rolling a 6 (P = 5/6 ≈ 0.83) |
Basic Probability Formula
Where:
- P(A)= Probability of event A occurring
- Favorable= Outcomes where A happens
- Total= All possible outcomes
Fundamental Probability Rules
Probability follows strict mathematical rules that govern how probabilities combine. These rules are essential for calculating complex probabilities from simple ones.
| Rule | Formula | When to Use |
|---|---|---|
| Complement Rule | P(not A) = 1 - P(A) | Finding probability of opposite event |
| Addition Rule (Mutually Exclusive) | P(A or B) = P(A) + P(B) | Events that can't both occur |
| Addition Rule (General) | P(A or B) = P(A) + P(B) - P(A and B) | Any two events |
| Multiplication Rule (Independent) | P(A and B) = P(A) × P(B) | One event doesn't affect the other |
| Multiplication Rule (Dependent) | P(A and B) = P(A) × P(B|A) | One event affects the other |
Key insight: The sum of probabilities of all possible outcomes must equal 1. This is why P(not A) = 1 - P(A).
Conditional Probability
Conditional probability is the probability of an event given that another event has already occurred. It's written as P(A|B), read as "probability of A given B." This concept is crucial for understanding dependent events and is the foundation of Bayes' theorem.
| Scenario | Unconditional P | Conditional P | Difference |
|---|---|---|---|
| Drawing a heart from deck | P(heart) = 13/52 = 25% | P(heart|red card) = 13/26 = 50% | Knowledge changes probability |
| Rain tomorrow | P(rain) = 30% | P(rain|cloudy today) = 60% | Prior information helps |
| Disease test positive | P(disease) = 1% | P(disease|positive test) = varies | Base rate matters (Bayes) |
Conditional Probability Formula
Where:
- P(A|B)= Probability of A given B occurred
- P(A and B)= Probability both A and B occur
- P(B)= Probability of B
Bayes' Theorem
Bayes' theorem lets you update probabilities based on new evidence. It's essential for medical diagnosis, spam filtering, machine learning, and rational decision-making. The theorem reverses conditional probabilities: if you know P(B|A), you can find P(A|B).
| Component | Name | Meaning |
|---|---|---|
| P(A|B) | Posterior probability | Updated probability after evidence |
| P(A) | Prior probability | Initial probability before evidence |
| P(B|A) | Likelihood | Probability of evidence if A is true |
| P(B) | Marginal likelihood | Total probability of evidence |
Bayes' Theorem
Where:
- P(A|B)= Posterior: probability of A given evidence B
- P(B|A)= Likelihood: probability of B if A is true
- P(A)= Prior: initial probability of A
- P(B)= Total probability of evidence B
Independent vs Dependent Events
Events are independent if one occurring doesn't affect the probability of the other. For independent events, P(A|B) = P(A). For dependent events, knowing one occurred changes the probability of the other.
| Type | Characteristic | Example | Multiplication Rule |
|---|---|---|---|
| Independent | One doesn't affect the other | Two coin flips | P(A and B) = P(A) × P(B) |
| Dependent | One affects the other | Cards without replacement | P(A and B) = P(A) × P(B|A) |
Testing independence: Events A and B are independent if and only if P(A and B) = P(A) × P(B).
Common mistake: Confusing independent with mutually exclusive. Mutually exclusive events (can't both happen) are actually dependent—if one happens, the other has probability 0.
Counting: Permutations and Combinations
Many probability problems require counting possible outcomes. Permutations count arrangements where order matters; combinations count selections where order doesn't matter.
| Concept | Formula | When to Use | Example |
|---|---|---|---|
| Permutation | P(n,r) = n!/(n-r)! | Order matters | Arranging 3 people in 3 seats: 6 ways |
| Combination | C(n,r) = n!/[r!(n-r)!] | Order doesn't matter | Choosing 3 from 5 people: 10 ways |
| Factorial | n! = n×(n-1)×...×1 | Arranging all n items | 5! = 120 |
Combination Formula (Choose)
Where:
- n= Total number of items
- r= Number of items to choose
- != Factorial (e.g., 5! = 120)
Common Probability Distributions
Probability distributions describe patterns of random outcomes. Understanding common distributions helps solve real-world probability problems.
| Distribution | Use Case | Example | Key Parameter |
|---|---|---|---|
| Uniform | Equal probability outcomes | Rolling a die | Min, max values |
| Binomial | Fixed trials, yes/no outcomes | Coin flips | n trials, p probability |
| Poisson | Events in time/space | Customer arrivals per hour | λ (average rate) |
| Normal | Continuous, bell-shaped | Heights, IQ scores | μ (mean), σ (SD) |
| Exponential | Time between events | Time until next customer | λ (rate) |
Worked Examples
Basic Probability: Drawing Cards
Problem:
What's the probability of drawing a King or a Heart from a standard 52-card deck?
Solution Steps:
- 1Count Kings: 4 cards
- 2Count Hearts: 13 cards
- 3Count King of Hearts (overlap): 1 card
- 4Apply addition rule: P(King or Heart) = P(King) + P(Heart) - P(King and Heart)
- 5Calculate: 4/52 + 13/52 - 1/52 = 16/52 = 4/13
Result:
P(King or Heart) = 16/52 ≈ 30.8%. We subtract the King of Hearts to avoid double-counting.
Conditional Probability: Medical Testing
Problem:
A disease affects 1% of the population. A test is 95% accurate for positive cases and 90% accurate for negative cases. If you test positive, what's the probability you have the disease?
Solution Steps:
- 1Prior: P(disease) = 0.01, P(no disease) = 0.99
- 2Likelihood: P(positive|disease) = 0.95, P(positive|no disease) = 0.10
- 3Total positive: P(positive) = 0.95×0.01 + 0.10×0.99 = 0.0095 + 0.099 = 0.1085
- 4Apply Bayes: P(disease|positive) = (0.95 × 0.01) / 0.1085
- 5Calculate: 0.0095 / 0.1085 = 0.0876
Result:
P(disease|positive) ≈ 8.76%. Despite a positive test, there's only ~9% chance of having the disease! This is because the disease is rare (base rate fallacy).
Independent Events: Multiple Trials
Problem:
What's the probability of getting at least one 6 when rolling a die 4 times?
Solution Steps:
- 1P(not 6 on one roll) = 5/6
- 2P(no 6 in 4 rolls) = (5/6)⁴ (independent events)
- 3Calculate: (5/6)⁴ = 625/1296 ≈ 0.482
- 4Apply complement: P(at least one 6) = 1 - P(no 6s)
- 5Calculate: 1 - 0.482 = 0.518
Result:
P(at least one 6) ≈ 51.8%. The complement rule often simplifies 'at least one' problems.
Tips & Best Practices
- ✓For 'at least one' problems, use the complement: P(at least one) = 1 - P(none).
- ✓Draw probability trees for complex conditional probability problems.
- ✓Check independence by testing if P(A and B) = P(A) × P(B).
- ✓Remember that mutually exclusive events (can't both happen) are NOT independent.
- ✓Bayes' theorem is powerful when you know P(evidence|hypothesis) but want P(hypothesis|evidence).
- ✓For counting problems: does order matter? If yes, permutations; if no, combinations.
- ✓The base rate (prior probability) dramatically affects posterior probability—don't ignore it!
Frequently Asked Questions
Sources & References
Last updated: 2026-01-22