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16 terms

Probability, Odds and the Counting Principle

Basic probability formulas and definitions
STUDY
PLAY
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
Permutation
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
n!
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!
Combination
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