Math 151 , Spring '07 Friday Day 24, March 30 Hit reload... After class Revised day 25, corrected day 28

Principles of designing an experiment: Compare groups with different treatments:   Control as much as you can, to make all the groups the same except for treatments, Randomize the rest; Use enough subjects  to average out bad "chance" .
   "Randomized comparative experiment"

"Probability" of particular something happening: proportion of times it would happen in a very long series of (independent) repetitions of the phenomenon. (Re)visit Monday:  Applet:  Probability
    (independence:  outcome of one trial (repetition) must not influence the outcome of any other.)
The random number table.  At each place, the probability of any particular one of the 10 digits is 1/10, or .10.     Sets of 25 digits  from the table iIndividual sets of 25 showed much variability.  Pooled  shows more "flatness" --but still much variability.  You would be right to be skeptical when I told you that your "pick-a-number" choices were not random, on the basis of just this class's data.  Not enough to necessarily show the pattern.

Got to here Friday No new material Monday.  Exam 3 stops here:
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)

Pick one person at random from U.S. Pop. (Age 25 +)   Probability = proportion in the population

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

P( that the person's education is HS or less) = ?
P(Not finished college or didn't start) = ?

Discrete models: (Can make a list of all members of the sample space)  Make the list, and
Assign a probability to each outcome (>0) so they add to 1.   (Sometimes equal values make sense.)
    Prob. of an event is sum of prob's of its outcomes.

Recall exercise 8.10:  Pick 6 people from list of 28 managers.  How many people of Asian surname do you get?  Excel analysis

(You won't learn how to calculate these probabilities.  Ch. 13, which we'll skip, gives a hint.  Numbers corrected 4/8)

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      |

Often the sample space is naturally expressed in numbers, thus
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."
Flip coin twice. R.V. X = number of heads:  Distribution given by table.
             x| 2  |    1    |  0 |
       P(X=x) | .25|   .50   | .25|  P(X > 1) = ?   Words:  Prob that # heads is > 1
                                     P(X = 2) = ?         Prob that # heads is 2