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- The Sample Space
is the
*set*of all possible*outcomes*from a random experiment. - 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) }*- Event is a subset of the sample Space
- 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) }*- Probability is afunction that maps events into the range [0,1]
- 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 - P(Ø) = 0
- probability of nothing happening is 0
- P(S) = 1
- The probability of something happening is 1
- P(A
*U*B) = P(A) + P(B) if A and B are mutually exclusive- probability is additive over distinct events
- probability of nothing happening is 0
- 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.

Document Last revised: 1997/1/5