Probability Guide
Classical Probability
P(A) = (favorable outcomes) / (total equally likely outcomes). Example: P(heads) = 1/2, P(rolling 6) = 1/6
Addition Rule
P(A โช B) = P(A) + P(B) โ P(A โฉ B)
If mutually exclusive: P(A โช B) = P(A) + P(B)
Multiplication Rule
P(A โฉ B) = P(A) ร P(B|A)
If independent: P(A โฉ B) = P(A) ร P(B)
Conditional Probability
P(A|B) = P(A โฉ B) / P(B) โ probability of A given B has occurred.
Bayes' Theorem
P(A|B) = P(B|A) ร P(A) / P(B)
Used to update belief based on new evidence. Foundation of Bayesian statistics and ML classifiers.
Expected Value
E(X) = ฮฃ xแตข ยท P(xแตข) โ the weighted average of all possible outcomes. Example: E(fair die) = (1+2+3+4+5+6)/6 = 3.5