Probability Calculator

Understanding Probability

Probability measures the likelihood of an event occurring, expressed as a value between 0 (impossible) and 1 (certain), or 0% to 100%.

Basic Probability Rules

1. Basic Probability

Example: Rolling a die, P(rolling a 3) = 1/6 ≈ 0.167 or 16.7%

2. Complement Rule

The probability an event doesn't occur is 1 minus the probability it does occur.

3. Addition Rule (OR)

For mutually exclusive events: P(A OR B) = P(A) + P(B)

Example: Drawing a heart OR a king from a deck

4. Multiplication Rule (AND)

For independent events: P(A AND B) = P(A) × P(B)

Example: Flipping heads AND rolling a 6

Conditional Probability

The probability of event A given that event B has occurred:

Example: Probability of rain given cloudy sky

Bayes' Theorem

Update probabilities based on new evidence:

Where:

• P(A|B): Posterior probability (what we want to find)
• P(B|A): Likelihood (probability of evidence given hypothesis)
• P(A): Prior probability (initial belief)
• P(B): Marginal probability of evidence

Full Bayes' Formula:

Key Concepts

Real-World Applications

Medical Testing (Bayes' Theorem)

Problem: Disease affects 1% of population. Test is 90% accurate (sensitivity). False positive rate is 5%.

Question: If you test positive, what's the probability you have the disease?

Solution: P(Disease|Positive) = (0.90 × 0.01) / [(0.90 × 0.01) + (0.05 × 0.99)] ≈ 0.154 or 15.4%

Other Applications

• Weather Forecasting: Probability of rain given cloud conditions
• Spam Filtering: Probability email is spam given certain words
• Quality Control: Probability of defect given inspection results
• Sports: Win probabilities based on current score/time
• Insurance: Risk assessment and premium calculation
• Machine Learning: Classification and prediction algorithms

Common Probability Distributions

• Uniform: All outcomes equally likely (fair die)
• Binomial: Number of successes in fixed trials
• Normal: Bell curve, continuous outcomes
• Poisson: Count of events in fixed interval