Queens University at Kingston


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A. Sample Space and Probability

The Sample Space is the set of all possible outcomes from a random experiment.


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


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

  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: $P(B | A) = {P(A,B) \over P(A)}$
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)

$$P(B | A) = {1/52\over 4/52} = {1\over 4}$$
In words: There is a 1/4 chance that a 3 is also a Spade
$$P( C | A ) = {0 \over 4/52} = 0$$
In words: There is no chance that any 3 is also the 8 of Hearts.

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Document Last revised: 1997/1/5