Calculate probability of two events, normal distribution probabilities, and solve complex probability problems. Includes union, intersection, conditional probability, and statistical analysis with detailed explanations.
Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probability theory is fundamental in statistics, science, finance, gambling, artificial intelligence, and many other fields.
Understanding probability helps us make informed decisions under uncertainty, analyze risks, interpret data, and solve complex problems involving chance and randomness.
For equally likely outcomes
Probability that event A does not occur
| Event | Formula | Description | Example |
|---|---|---|---|
| P(A ∪ B) | P(A) + P(B) - P(A ∩ B) | A or B or both | Drawing red card or ace |
| P(A ∩ B) | P(A) × P(B) if independent | Both A and B | Two heads in coin tosses |
| P(A|B) | P(A ∩ B) / P(B) | A given B occurred | Rain given cloudy sky |
| P(A') | 1 - P(A) | Not A | Not rolling a six |
| P((A ∪ B)') | 1 - P(A ∪ B) | Neither A nor B | Neither red nor ace |
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about the mean. It is characterized by its bell-shaped curve and is fundamental in statistics.
Where μ is mean and σ is standard deviation
| Type | Definition | Example | Calculation |
|---|---|---|---|
| Classical | Equally likely outcomes | Coin flip, dice roll | Favorable/Total outcomes |
| Empirical | Based on observed data | Weather forecasting | Observed frequency |
| Subjective | Personal judgment | Stock market prediction | Expert opinion |
| Conditional | Given another event | P(Rain|Cloudy) | P(A∩B)/P(B) |
| Distribution | Type | Use Case | Parameters |
|---|---|---|---|
| Normal | Continuous | Heights, weights, test scores | μ (mean), σ (std dev) |
| Binomial | Discrete | Success/failure trials | n (trials), p (success probability) |
| Poisson | Discrete | Rare events, arrivals | λ (rate parameter) |
| Uniform | Continuous | Random selection | a (min), b (max) |
| Exponential | Continuous | Waiting times | λ (rate parameter) |
Addition Rule: For any two events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Multiplication Rule: For independent events, P(A ∩ B) = P(A) × P(B)
Law of Total Probability: P(A) = P(A|B₁)P(B₁) + P(A|B₂)P(B₂) + ... + P(A|Bₙ)P(Bₙ)
Bayes' Theorem: P(A|B) = P(B|A) × P(A) / P(B)
| Field | Application | Example | Probability Concept |
|---|---|---|---|
| Medicine | Diagnostic testing | Disease screening accuracy | Conditional probability |
| Finance | Risk assessment | Investment portfolio risk | Normal distribution |
| Insurance | Premium calculation | Accident probability | Empirical probability |
| Quality Control | Defect rates | Manufacturing quality | Binomial distribution |
| Weather | Forecasting | Precipitation chance | Statistical models |
Identify the Sample Space: Clearly define all possible outcomes before calculating probabilities.
Check for Independence: Determine whether events are independent or dependent before applying formulas.
Use Complement When Easier: Sometimes it's easier to calculate P(A') and subtract from 1.
Draw Diagrams: Use tree diagrams, Venn diagrams, or probability tables to visualize problems.