Probability
Essential formulas and concepts in probability
Probability (classical definition)
Variables:
- \(A is an event in a finite sample space with equally likely outcomes.\)
Description/Usage:
Gives the probability of event \(A\) by ratio if all outcomes are equally likely. For example, the probability of rolling a 5 on a fair six-sided die is \(1/6\).
Complement Rule
Variables:
Description/Usage:
The probability that \(A\)
Addition Rule (Union of Events)
Variables:
Description/Usage:
Conditional Probability
Variables:
Description/Usage:
Updates the probability of \(A\) under the condition that \(B\) is known to have happened. For example, the probability it’s raining given that the sky is cloudy. Visually explained using a restricted sample space (just \(B\) outcomes).
Bayes’ Theorem
Variables:
Description/Usage:
Allows inversion of conditional probabilities. It’s fundamental in statistical inference, relating the probability of a cause \(A\) given effect \(B\) to the probability of effect given cause. Often visualized with probability trees or Bayes’ boxes.
Binomial Distribution (Probability of $k$ successes)
Variables:
Description/Usage:
Gives the probability of exactly \(k\) successes in \(n\) Bernoulli trials. For example, the chance of getting exactly 3 heads in 5 fair coin flips. The distribution’s probabilities can be visualized with a histogram or Pascal’s triangle (for small \(n\)).
Expected Value (Mean) of a Discrete Random Variable
Variables:
- \(X is a random variable that takes values x_i with probability P(X=x_i).\)