## A. Sample Space and Probability

#### Example

the random experiment of drawing one card from a deck has a sample space with 52 elements (outcomes):
S = { (2,Diamonds), ...., (Ace,Diamonds), (2,Hearts), ...., (Ace,Hearts), (2,Clubs), ..., (Ace,Clubs), (2,Spades), ..., (Ace,Spades) }

#### Example

the event of drawing a three is
A = { (3,Diamonds), (3,Hearts), (3,Clubs), (3,Spades) }
An event can be a simple outcome:
B = { (3,Spades) }
C = { (8,Hearts) }

#### Example (continued from above)

in a fair deck
P(A) = 4/52 P(B) = 1/52 P(A & C) = 0 (A and C can't occur simultaneously)

#### Some laws of Probability

1. P(Ø) = 0
probability of nothing happening is 0
2. P(S) = 1
The probability of something happening is 1
3. P(A U B) = P(A) + P(B) if A and B are mutually exclusive
probability is additive over distinct events

#### Events A and B are independent if and only if P(A & B) = P(A)*P(B) Conditional Probability: P( B | A) = probability that event B occurs given (conditional) that event A has occurred. Bayes Theorem: In words: given A has occurred, the probability that B also occurs is the probability that they both occur divided by the probability that A occurs. Example (using events A, B, and C defined above) In words: There is a 1/4 chance that a 3 is also a Spade In words: There is no chance that any 3 is also the 8 of Hearts.

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