Math 151 , Fall 2007 Monday Day 25, Oct. 22 Hit reload...After class.

Probability Models : (p. 250-256)
   Random phenomenon, described by
    Sample space S:  set of all possible outcomes (no overlap of descriptions)
    Event:  any outcome or set of outcomes
    Probability model: S, and a way of assigning a probability to each event.
Sample space depends on what you want to know:
Phenomenon: Flip coin twice.
    S₁ = {HH, HT, TH, TT}     S₂ = {0, 1, 2} number of heads   S₃ = {Y, N} both are heads?
  ..
We looked at the probabilities for these, implicitly using the "common sense" rules for proportions just below.

Probability rules:  pp. 253, in words, then in notation.
A an event in sample space S, P(A) is "the probability that  A occurs"
    These rules are all true for proportions in long run (Probabilities), proportion of counts, proportions of areas.
    1.  0 < P(A) < 1   (any probability is a number between 0 and 1. )
    2. P(S) = 1         (all the outcomes together have total probability 1)
    3.  A and B are  disjoint if they have no outcomes in common (can't happen simultaneously.)
        If A and B are disjoint, their probabilities add:  P(A or B) = P(A) + P(B)
    4. For any event A, P(A does not occur) = 1 - P(A)

Discrete models: Assign a probability to each outcome (>0) so they add to 1.  Prob. of an event is sum of prob's of its outcomes.
 Start with Random Variable next time:
Random Variable:  (X, Y, Z...) Variable whose value is a numerical outcome of a random phenomenon.
 Probability distribution of X tells us what values X can take and how to assign probabilities to them.
    If X has a finite number of possible values (Discrete distributions), nothing new except notation.
     P(X < 2) is "Prob. that X is less than 2."
New/recap:
Probabilities follow the "common sense" rule for proportions of a whole.  Same rules for proportions of areas, proportions of counts, proportions in histograms, proportions of times in the long run something would happen. 
Two principles for assigning probabilities:
--Sometimes, a properly chosen sample space will have equally likely outcomes. You can use this to find other probabilities.
--If you pick an individual at random from a population, the probability that one individual will be XYZ is the same as the proportion of XYZ's in the population.
Next:   Continuous Sample Spaces.