Math 151 , Spring 2006, Day 25 Mon. April 3 Hit reload ...After class

We know that a sample from a population will not exactly represent the population.  If we take a random sample, the behavior of samples will not be individually predictable, but there will be predictable pattern in many random samples from the same population.  Knowing the pattern will be  as good as we can do.  Need probability.
Recall (Day20):p. 227      Sample Chosen from a  Population
  Numerical summary: Statistic (Latin)  Parameter(Greek letter)

Chance  behavior (a random phenomenon): Unpredictable in the short run,  predictable regular pattern in the long run.
(Prof. Persi Diaconis (a table magician) can flip a coin so precisely it always comes up the way he wants.His coinflipping is not a random phenomenon.  Mine is.
"Probability" of particular something happening: proportion of times it would happen in a very long series of independent repetitions (trials) of the phenomenon: "long-run relative frequency".
    (independence:  outcome of one trial must not influence the outcome of any other.)

Law of Large Numbers (LLN):  Relative frequency of repeated independent trials gets closer  to the "true" relative frequency as the number of trials increases.
  (But it may take a long time: Large Numbers of trials. Use  http://www.whfreeman.com/scc -- "Probability " 1 toss at a time--settles down slowly.   )
(&&Another version of  LLN says the mean from a sample of size n gets closer and closer to the true = "population" mean, as you take bigger samples (as n increases).  Activstats presents this, 14-1, and we'll return to this soon.)

Aberrations won't be compensated for; they will only be swamped out.  (Misconception of "law of averages.")  Lady luck never owes you anything!

Start here Wed:
Probability Model:
A Random phenomenon,
    Sample space S:  set of all possible outcomes (no overlap of descriptions) (def. p. 284)
    Event:  any  set of outcomes(including one outcome, & even the set containing no 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?

Probability rules:  (pp. 274-6, 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), prop.of counts, proportions of areas.
    1.  0 < P(A) < 1
    2. P(S) = 1
    3. For any event A, P(A does not occur) = 1 - P(A)
    4.  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)

Pick one person from U.S. Pop. (Age 25 +)

Sample space:	No HS degree	HS only     .	1-3 yrs College	4 + yrs College
Proportion in pop.	18.3%	33.9%	24.8%	23.0%
Probability	.183	.339	.248	.230

P(No 4-year "degree") = ?
P( HS or less) = ?

Finite sample spaces (you can list the outcomes):
Assign a probability to each outcome (>0) so they add to 1.   (Sometimes equal values--"equally likely" make sense.)
    Prob. of an event is sum of prob's of its outcomes.

Phenomenon: Flip coin twice.
    S₁ = {HH, HT, TH, TT}     S₂ = {0, 1, 2} number of heads   S₃ = {Y, N} both are heads?
Sample space  | HH | HT | TH | TT |
       Prob's | .25| .25| .25| .25|  P(tail followed by head)=?
Sample space  | 2  |    1    |  0 |  P(at least 1 tail)=?   P(1 of each) = ?
       Prob's | .25|   .50   | .25|  P(at least 1 Head)= ?  P(2 Heads) = ?
Sample space  | Y  |       N      |
       Prob's | .25|     .75      |