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Basic probability formulas and definitions

Odds against event E

P (fail) ÷ P (success)
Alternate notation P(fail) : P (success)

Expected value

P₁A₁ + P₂A₂ + ... + PnAn for events 1, 2, ... n
where Pn is the probability that n occurs and An is the value or cost if it does occur

Sample space

List of all possible outcomes of an event
Tree diagrams are helpful in determining sample space

Counting principle

The probability of two experiments are multiplied to determine the overall probability. Ex: a coin toss (2 outcomes) followed by the roll of a die (6 outcomes) has 2×6=12 outcomes (H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6)

Independent events

Occurrence of either event does not affect probability of the other
Ex: rolling a die or tossing a coin multiple times (with replacement)

Dependent events

Occurrence of one event depends on outcome of the other
Ex: drawing repeatedly from a deck of cards (without replacement)

Addition formula ("OR" problems)

P (A or B) = P (A) + P (B) - P (A and B)
Easy to see this with Venn diagram; subtracting the 3rd term prevents counting the "middles" twice


Any ordered arrangement of a set of objects where order matters
Ex: abc, acb, bac, bca, cab, cba are all the permutaions of a,b,c

Number of permutations of n distinct items


Number of permutations of n objects if some are identical

n! / (n₁! n₂! ... nr!) where n₁, n₂... are groupings of identical items
Ex: bag of 9 balls w/ 3 red, 3 yellow, and 3 blue has 9! / 3! 3! 3!


An arrangement of a set of objects where order does not matter
Ex: abc and cba are not distinct but the same combination of a,b,c

Number of permutations for selecting r items from among n

nPr = n! / (n-r)!
Ex: {a,b,c} select 2 from 3 → 3! / (3-2)! = 6 (ab,ac,ba,bc,ca,cb)

Number of combinations for selecting r items from among n

nCr = n! / r! (n-r)!
Ex: {a,b,c} select 2 from 3→ 3! / 2! (3-2)! = 3 (ab,ac,bc)

Multiplication of probabilities ("AND" problems)

P (A and B) = P(A) × P(B) , assuming event A has occurred
Alternate notation: P (B | A), probability of B given A

Find P(B | A)

A has already occurred, so that is the new sample space (denominator) in a probability calculation

Card dealing - Ex: probability of getting exactly three 6's in a five card hand

ways of choosing three 6's * ways of choosing two non-6's ÷ ways of choosing 5 from 52 = ₄C₃ * ₄₈C₂ / ₅₂C₅

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