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Basic probability formulas and definitions

### 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 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