Probability & Combinatorics Calculator
Calculate permutations, combinations, and probability with detailed solutions
Probability & Combinatorics
How to Use the Probability & Combinatorics Calculator
Key Concepts
- Permutation: Arrangement where order matters (P(n,r) = n!/(n-r)!)
- Combination: Selection where order doesn't matter (C(n,r) = n!/(r!(n-r)!))
- Factorial: Product of all positive integers up to n (n!)
- Probability: Likelihood of an event (0 ≤ P ≤ 1)
- Binomial: Probability of k successes in n independent trials
When to Use Each Type
- Permutations: Arranging people in line, race finishing orders
- Combinations: Choosing team members, lottery number selection
- Basic Probability: Dice rolls, card draws, simple events
- Binomial Probability: Coin flips, success rates, quality control
Important Rules
- For Permutations/Combinations: 0 ≤ r ≤ n
- For Probability: 0 ≤ P(E) ≤ 1
- For Binomial: Each trial must be independent
- Factorial Special Cases: 0! = 1, 1! = 1
Applications
- Statistics: Hypothesis testing, confidence intervals
- Gaming: Odds calculation, strategy development
- Business: Risk analysis, quality control
- Science: Experimental design, data analysis
Frequently Asked Questions
What's the difference between permutations and combinations?
Permutations count arrangements where order matters (like race positions), while combinations count selections where order doesn't matter (like choosing team members).
When do I use binomial probability?
Use binomial probability when you have a fixed number of independent trials, each with the same probability of success. Examples include coin flips, pass/fail tests, or quality control scenarios.
Why does 0! equal 1?
By mathematical convention and definition, 0! = 1. This makes many formulas work correctly, particularly in combinatorics and calculus.
How do I interpret probability values?
Probability ranges from 0 (impossible) to 1 (certain). Values near 0 indicate unlikely events, while values near 1 indicate likely events. 0.5 represents equal likelihood.
What are the limitations of these calculations?
These formulas assume independence between events and uniform probability distributions. Real-world scenarios may have dependencies or varying probabilities that require more complex analysis.
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