Math 151 , Fall 2002, Wednesday Day 23, Oct. 23 Hit reload to get most current version

Questions for HW?  exam?
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Ch. 4, Probability and Sampling Distributions.
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.

Sec. 4.1:  Sample/Population see day 22

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Chance  behavior (a random phenomenon): Unpredictable in the short run,  predictable regular pattern in the long run.
    Random numbers:  equally likely in the long run.
   "Random" here is more general--pattern is not necessarily equally likely

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 of the phenomenon.
    (independence:  outcome of one trial (repetition) must not influence the outcome of any other.)

Sec. 4.2 Probability Models
Random phenomenon,
    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?

Probability rules:  pp. 222-3, 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(not finished college) = ?
P( HS or less) = ?

Finite sample spaces:
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.

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(2Heads) = ?
Sample space  | Y  |       N      |
       Prob's | .25|     .75      |

Start here Monday:  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, 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)= ?  P(X = 2) = ?