Probability: A number from 0 to 1 telling how likely an event is
Impossible event: Probability 0
Certain event: Probability 1
Outcome: One possible result of an experiment
Event: A set of outcomes
Sample space: The set of all possible outcomes
Experiment (trial): An action with an uncertain result, like rolling a die
Theoretical probability: Favorable outcomes divided by all equally likely outcomes
Experimental probability: Number of times it happened divided by number of trials
Complement of an event: All outcomes where the event does not happen; P(not A) = 1 − P(A)
Independent events: One event does not change the probability of the other
Dependent events: One event changes the probability of the other
Mutually exclusive events: Events that cannot happen at the same time
Multiplication rule (independent): P(A and B) = P(A) × P(B)
Addition rule: P(A or B) = P(A) + P(B) − P(A and B)
Conditional probability: P(A | B) = P(A and B) / P(B)
Bayes' theorem: P(A | B) = P(B | A) P(A) / P(B)
Law of large numbers: With many trials, experimental probability gets close to theoretical probability
Expected value: The long-run average: sum of each value times its probability
Random variable: A variable whose value comes from a random process
Probability of heads on a fair coin: 1/2
Probability of a 4 on a fair die: 1/6
Probability of an even number on a die: 1/2
Probability of a heart from a standard deck: 13/52 = 1/4
Probability of an ace from a standard deck: 4/52 = 1/13
Cards in a standard deck: 52
Probability of two heads in two fair flips: 1/4
Probability of a sum of 7 with two dice: 6/36 = 1/6
Number of outcomes when rolling two dice: 36
Number of outcomes in 3 coin flips: 8
Fundamental counting principle: Multiply the number of choices at each step
Outfits from 3 shirts and 4 pants: 12
Factorial n!: n × (n − 1) × ... × 1
Permutation: An ordered arrangement
Permutation formula: P(n, r) = n! / (n − r)!
Combination: A selection where order does not matter
Combination formula: C(n, r) = n! / (r!(n − r)!)
Ways to arrange 5 books on a shelf: 5! = 120
P(5, 2): 20
C(5, 2): 10
C(6, 3): 20
C(10, 2): 45
C(n, 0) and C(n, n): Both equal 1
Pascal's triangle: A triangle where each number is the sum of the two above it; rows give combinations
Binomial distribution: The distribution of the number of successes in a fixed number of independent yes/no trials
With replacement: Each item is put back before the next draw
Without replacement: Items are not put back; later probabilities change
Tree diagram: A branching picture that lists outcomes of a multi-step event
Venn diagram: Overlapping circles showing how sets share members
Odds: Ratio of favorable outcomes to unfavorable outcomes
Fair game: A game where each player has the same expected value
Simulation: Using random numbers or objects to imitate a real chance process

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