Probability Calculator
Compute probabilities for various events and distributions easily. Free Probability Calculator for students and researchers.
Basic Probability
P(E) = Favorable Outcomes / Total Outcomes
Result
Probability Calculation
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* For educational purposes only
What is Probability?
Probability is a mathematical measure of how likely an event is to occur. It's expressed as a number between 0 and 1:
- 0 = Impossible (will never happen)
- 0.5 = Equal chance (50/50)
- 1 = Certain (will definitely happen)
Probability is fundamental to statistics, gambling, risk assessment, and decision-making under uncertainty.
Basic Probability Formula
P(E) = Number of Favorable Outcomes / Total Number of Outcomes
Example: What's the probability of rolling a 4 on a standard die?
- Favorable outcomes: 1 (only one side shows 4)
- Total outcomes: 6 (six sides total)
- P(rolling 4) = 1/6 ≈ 0.167 or 16.7%
Combinations vs Permutations
Combinations (nCr)
When order DOESN'T matter
Formula: C(n,r) = n! / (r! × (n-r)!)
Example: Choosing 3 students from a class of 10 for a committee.
- C(10,3) = 10! / (3! × 7!) = 120 ways
- Order doesn't matter: {Alice, Bob, Carol} = {Carol, Alice, Bob}
Real-world uses:
- Lottery numbers
- Committee selection
- Card hands in poker
Permutations (nPr)
When order DOES matter
Formula: P(n,r) = n! / (n-r)!
Example: Assigning 1st, 2nd, 3rd place from 10 competitors.
- P(10,3) = 10! / 7! = 720 ways
- Order matters: (Alice 1st, Bob 2nd, Carol 3rd) ≠ (Carol 1st, Alice 2nd, Bob 3rd)
Real-world uses:
- Race positions
- Password permutations
- Task scheduling
Key Difference: nPr ≥ nCr (permutations are always greater or equal to combinations for the same n and r)
Conditional Probability
P(A|B) reads as "probability of A given B" — the probability that A occurs, knowing that B has already occurred.
Formula: P(A|B) = P(A ∩ B) / P(B)
Example: In a deck of cards:
- P(Card is King | Card is a face card)
- P(King ∩ Face card) = 4/52 (4 kings)
- P(Face card) = 12/52 (J, Q, K in 4 suits)
- P(King | Face) = (4/52) / (12/52) = 4/12 = 1/3
Applications:
- Medical testing (probability of disease given positive test)
- Weather forecasting
- Machine learning predictions
Probability Rules
Addition Rule (OR)
P(A or B) = P(A) + P(B) - P(A and B)
For mutually exclusive events (can't both happen):
- P(A or B) = P(A) + P(B)
- Example: P(rolling 1 or 2) = 1/6 + 1/6 = 2/6 = 1/3
Multiplication Rule (AND)
P(A and B) = P(A) × P(B|A)
For independent events (one doesn't affect the other):
- P(A and B) = P(A) × P(B)
- Example: P(heads on coin AND rolling 6) = 1/2 × 1/6 = 1/12
Complement Rule
P(A') = 1 - P(A)
The probability of "not A" equals 1 minus the probability of A.
- Example: P(not rolling 6) = 1 - 1/6 = 5/6
Examples
Example 1: Basic Probability
Q: A bag contains 5 red, 3 blue, and 2 green marbles. What's the probability of drawing a blue marble?
Solution:
- Favorable outcomes: 3 (blue marbles)
- Total outcomes: 10 (all marbles)
- P(blue) = 3/10 = 0.3 or 30%
Example 2: Combinations
Q: How many 5-card poker hands can be dealt from a 52-card deck?
Solution:
- n = 52, r = 5
- C(52,5) = 52! / (5! × 47!) = 2,598,960 hands
Example 3: Permutations
Q: How many 4-digit PINs can be formed using digits 0-9 without repetition?
Solution:
- n = 10, r = 4
- P(10,4) = 10! / 6! = 10 × 9 × 8 × 7 = 5,040 PINs
Example 4: Conditional Probability
Q: 60% of students study. Of those who study, 80% pass. Of those who don't, 30% pass. If a student passed, what's the probability they studied?
Solution (using Bayes' Theorem):
- P(Study) = 0.6, P(Pass|Study) = 0.8
- P(Don't Study) = 0.4, P(Pass|Don't) = 0.3
- P(Pass) = 0.6×0.8 + 0.4×0.3 = 0.48 + 0.12 = 0.6
- P(Study|Pass) = P(Pass|Study) × P(Study) / P(Pass)
- = (0.8 × 0.6) / 0.6 = 0.48 / 0.6 = 0.8 or 80%
Applications of Probability
1. Risk Assessment
- Insurance premiums based on accident probability
- Investment risk vs. return analysis
- Medical treatment success rates
2. Games and Gambling
- Casino game odds
- Lottery probabilities
- Poker hand rankings
3. Quality Control
- Defect rates in manufacturing
- Sample testing reliability
- Process improvement
4. Data Science & AI
- Machine learning predictions
- Bayesian inference
- Statistical significance testing
Frequently Asked Questions
What's the difference between combinations and permutations?
Combinations (nCr): Used when order doesn't matter. Choosing a team, selecting lottery numbers. Permutations (nPr): Used when order matters. Arranging books on a shelf, assigning podium positions. Remember: nPr is always ≥ nCr for the same n and r.
Can probability be greater than 1?
No. Probability is always between 0 and 1 (inclusive). A value of 0 means impossible, 1 means certain. If you get a value >1, there's an error in your calculation.
What does P(A|B) mean?
P(A|B) is conditional probability — the probability of A happening given that B has already happened. Read as "P of A given B." It's calculated as P(A∩B) / P(B), where P(A∩B) is the probability of both A and B occurring.
When do I use the addition rule vs multiplication rule?
Use addition for "OR" questions (either A or B can happen): P(A or B). Use multiplication for "AND" questions (both A and B must happen): P(A and B). Be careful: addition rule subtracts P(A and B) to avoid double-counting unless events are mutually exclusive.
How do I calculate factorial for large numbers?
Factorials grow extremely fast (10! = 3.6 million, 20! ≈ 2.4 × 10¹⁸). For large n, use:
- Stirling's approximation: n! ≈ √(2πn) × (n/e)ⁿ
- Scientific calculator or programming
- Simplification: C(100,98) = C(100,2) = 4,950 (use smaller numbers)
What's the difference between theoretical and experimental probability?
Theoretical probability: What math predicts (coin flip = 0.5). Experimental probability: What actually happens in trials (flip 100 times, get 47 heads = 0.47). With more trials, experimental approaches theoretical (Law of Large Numbers).
References
This calculator uses fundamental probability theory based on standard mathematical principles: