Calculate probabilities for single events, multiple events, series, normal distributions, and Bayes theorem. Get real-time results with comprehensive analysis and detailed explanations.
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Single events, two events, series, normal distribution, Bayes, combinations, Poisson, and binomial
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Probability is a measure of the likelihood that an event will occur. It's expressed as a number between 0 and 1, where 0 means the event will never occur and 1 means it will always occur.
P(Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes
This represents the percentage likelihood of an event occurring
Events where the outcome of one does not affect the outcome of another.
P(A and B) = P(A) × P(B)
Example: Rolling two dice - the result of the first die doesn't affect the second.
Events where the outcome of one affects the probability of another.
P(A and B) = P(A) × P(B|A)
Example: Drawing cards without replacement - each draw affects subsequent probabilities.
Events that cannot occur simultaneously.
P(A or B) = P(A) + P(B)
Example: Rolling a die - getting a 3 and a 5 in a single roll is impossible.
Bayes' theorem describes the probability of an event based on prior knowledge of conditions related to the event, widely used in statistical analysis.
P(A|B) = P(B|A) × P(A) / P(B)
A continuous probability distribution that forms the famous bell curve.
f(x) = (1/σ√(2π)) × e^(-½((x-μ)/σ)²)
Describes the number of successes in a fixed number of independent trials.
P(X=k) = C(n,k) × p^k × (1-p)^(n-k)
Discrete probability distributions deal with countable outcomes, where each possible value has a specific probability. These distributions are fundamental in analyzing scenarios with finite or countably infinite outcomes.
Models the number of successes in a fixed number of independent trials, each with the same probability of success.
Models the number of events occurring in a fixed interval of time or space, given a known constant mean rate.
Models the number of trials needed to get the first success in a series of independent Bernoulli trials.
Continuous distributions model variables that can take any value within a range. The probability of any exact value is zero, but we can calculate probabilities for ranges of values.
The most important continuous distribution, characterized by its bell-shaped curve. Many natural phenomena follow this pattern.
Models the time between events in a Poisson process, where events occur continuously and independently.
Hypothesis testing is a statistical method for making decisions about population parameters based on sample data.
Statement of no effect or no difference
Statement we want to prove
Probability of Type I error (typically 0.05)
Reject H₀ if p-value < α
| H₀ True | H₀ False | |
|---|---|---|
| Reject H₀ | Type I Error (α) | Correct |
| Fail to Reject H₀ | Correct | Type II Error (β) |
One of the most fundamental theorems in statistics, the CLT explains why the normal distribution is so prevalent in nature and forms the foundation for many statistical inference procedures.
For a population with mean μ and standard deviation σ, the sampling distribution of sample means approaches a normal distribution as the sample size n increases, regardless of the shape of the original population distribution.
Probability theory forms the mathematical foundation for many machine learning algorithms, enabling systems to reason under uncertainty and make informed predictions.
Uses Bayes' theorem with "naive" independence assumption between features.
Models probability of binary outcomes using the logistic function.
Incorporate uncertainty in neural network weights using probability distributions.
Modern AI systems need to express confidence in their predictions and handle uncertain information effectively.
Irreducible uncertainty due to inherent randomness in the data or process.
Reducible uncertainty due to lack of knowledge, can be reduced with more data.
Bayesian statistics provides a principled way to incorporate prior knowledge and update beliefs as new evidence becomes available. This approach is fundamental to modern data science and artificial intelligence.
| Aspect | Frequentist | Bayesian |
|---|---|---|
| Parameters | Fixed but unknown constants | Random variables with distributions |
| Probability | Long-run frequency | Degree of belief |
| Inference | P-values, confidence intervals | Posterior distributions, credible intervals |
| Prior Knowledge | Not formally incorporated | Explicitly modeled through priors |
The finance industry heavily relies on probability theory for risk assessment, portfolio optimization, derivative pricing, and regulatory compliance.
Measures potential loss in portfolio value over a specific time period at a given confidence level.
Example: 95% VaR of $1M means 5% chance of losing more than $1M
Uses logistic regression and other probabilistic models to estimate default probability.
Black-Scholes model uses geometric Brownian motion to model stock prices.
Where S = stock price, μ = drift, σ = volatility, dW = Wiener process
Life tables and survival analysis for premium calculation.
Sensitivity, specificity, and positive predictive value calculations.
Disease spread modeling and public health decision making.
Risk prediction models for individual patients.
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