Probability Calculator
Calculate probabilities for independent events, combinations, permutations, and binomial distributions. Essential for statistics, gaming, and risk assessment.
Calculate Probability
Select a calculation type and enter the required values:
Simple: P = favorable/total
Combinations: C(n,k) = n!/(k!(n-k)!)
Permutations: P(n,k) = n!/(n-k)!)
Binomial: P(X=k) = C(n,k) ร p^k ร (1-p)^(n-k)
Understanding Probability
Probability is the branch of mathematics that deals with the likelihood of events occurring. It provides a mathematical framework for analyzing random phenomena and making informed decisions under uncertainty.
Types of Probability Calculations
๐ฏ Simple Probability
Basic probability calculation for equally likely outcomes.
Formula: P = favorable outcomes / total outcomes
Example: Probability of rolling a 6 on a die = 1/6
๐ข Combinations
Number of ways to choose items without regard to order.
Formula: C(n,k) = n! / (k! ร (n-k)!)
Example: Lottery combinations, committee selection
๐ Permutations
Number of ways to arrange items with regard to order.
Formula: P(n,k) = n! / (n-k)!
Example: Race finishing orders, password arrangements
Binomial Probability Distribution
๐ Binomial Distribution
Probability of exactly k successes in n independent trials.
Each trial has two possible outcomes (success/failure).
Success probability is constant for all trials.
๐ Probability Mass Function
P(X = k) = C(n,k) ร p^k ร (1-p)^(n-k)
Where: n = trials, k = successes, p = success probability
Mean = n ร p, Variance = n ร p ร (1-p)
๐ฒ Applications
Coin flips, quality control, opinion polls.
Drug trial success rates, election predictions.
Any scenario with success/failure outcomes.
Common Probability Scenarios
| Scenario | Type | Calculation | Probability |
|---|---|---|---|
| Fair coin toss (heads) | Simple | 1 favorable / 2 total | 50% or 0.5 |
| Six-sided die (rolling 4) | Simple | 1 favorable / 6 total | 16.67% or 1/6 |
| Lottery (6/49) | Combinations | C(49,6) = 13,983,816 | 1 in 13,983,816 |
| Password combinations | Permutations | P(26,8) for 8-letter passwords | 62,990,928,000 possibilities |
| Basketball free throws | Binomial | P(X=9) for 10 shots, p=0.8 | 26.84% |
Probability Rules and Theorems
โ Addition Rule
For mutually exclusive events: P(A โช B) = P(A) + P(B)
For non-exclusive events: P(A โช B) = P(A) + P(B) - P(A โฉ B)
Used when events cannot occur simultaneously
โ๏ธ Multiplication Rule
For independent events: P(A โฉ B) = P(A) ร P(B)
For dependent events: P(A โฉ B) = P(A) ร P(B|A)
Used when events occur together
๐ Complement Rule
P(not A) = 1 - P(A)
The probability of an event not occurring
Useful for calculating "at least one" scenarios
Practical Applications
๐ฒ Gaming & Gambling
Poker hand probabilities, lottery odds calculation.
Casino game analysis, sports betting probabilities.
Understanding house edges and expected values.
๐ Statistics & Research
Survey sampling methods, clinical trial analysis.
Quality control testing, opinion poll accuracy.
Market research and A/B testing analysis.
๐ผ Business & Finance
Risk assessment models, investment probability analysis.
Insurance premium calculations, project success prediction.
Customer behavior analysis and decision making.
๐ฌ Science & Engineering
Reliability engineering, failure rate calculations.
Quality assurance testing, experimental design.
Statistical process control and Six Sigma analysis.
Understanding Factorials
๐ข Factorial Definition
n! = n ร (n-1) ร (n-2) ร ... ร 1
0! = 1 by convention
Represents number of ways to arrange n distinct items
๐ Factorial Growth
Factorials grow extremely rapidly
5! = 120, 10! = 3,628,800
20! has 19 digits, 100! has 158 digits
Limits practical calculations for large n
๐งฎ Applications
Permutations and combinations calculations
Probability theory and statistics
Mathematical series and approximations
Computer science algorithms
Probability Distributions
๐ Discrete Distributions
Probability mass function for countable outcomes
Binomial, Poisson, geometric distributions
Used for counting scenarios
๐ Continuous Distributions
Probability density function for continuous variables
Normal, exponential, uniform distributions
Used for measurements and timing
๐ฏ Expected Value
Mean or average outcome of a random variable
E[X] = ฮฃ(x ร P(X=x)) for discrete variables
Long-run average over many trials
๐ก Probability Tip: When comparing probabilities, always consider the context. A 10% probability might seem small, but if the consequences are severe (like in safety or medical scenarios), it could be very significant. Always consider both the probability and the impact of the event.