Basic Probability 
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Def: A (finite) *probability space* is a finite set \Omega 
together with a function \mu: \Omega -> [0,1] having the property
that 
        \sum         \mu(x)   =   1
    x \in \Omega

We call \Omega the "sample space" and we call \mu the 
"probability measure".


Eg 1: You roll a fair die six times:
        \Omega = {1,2,3,4,5,6}
        \mu(1)=\mu(2)=...=\mu(6)=1/6

Eg 2: You deal out five cards from a deck of cards:
        \Omega = { 5-element subsets of the 52 cards }
        \mu(x) = 1 / C(52,5)  for all x \in }omega

Eg 3: You flip an unfair coin 10 times.  The coin lands heads 
      a fraction p=0.51 of the time:
         \Omega = {0,1}^10
         \mu(x) = p^{#1(x)} (1-p)^{#0(x)}    where #1(x) = the number of 1-bits
                                             in the string x and #0(x) = the number
                                             of 0-bits in the string x.


Def: Let (\Omega, \mu) be a probability space.  
     An *event* A is a subset of \Omega.


Def: Let A be an event of probability space (\Omega, \mu).
     Pr[A] =   \sum   \mu(x)
             x \in A
     "The probability of event A"


Eg:  Returning to Eg 1, "you roll an even number" is an event -- the 
     event is A = {2,4,6}.   Pr[A] = 3 * (1/6) = 1/2.

In general, whenever you hear "probability" make sure that you are clear
WHAT is the probability space and WHAT is the event in question.

Propositions:
      - Pr[\emptyset] = 0   // by definition 
      - Pr[\Omega] = 1
      - Pr[A] + Pr[\Omega -A] = 1
      - If A and B are disjoint events (that is, disjoint sets) then
        Pr[A u B] = Pr[A] + Pr[B]
      - ("sum bound")
        Pr[A u B] <= Pr[A] + Pr[B]
      - In general,
        Pr[A u B] = Pr[A] + Pr[B] - Pr[A intersect B]   // inclusion-exclusion principle