MATH 251, Probability and Statistics I, Fall 2005, Oct. 14, Day 21

Project due Monday:  Handout. p.2 is checklist
Read  4.1, 4.2 for this assignment.  I am assuming this is familiar to everyone,  so I've given a small amount of HW, and optional problems.  Note, the text uses (A and B) where you may be used to (A intersect B), (A or B) where you may be used to (A union B). For next class, read 4.3, Random variables.

Hand in: 
Sec. 4.1, p. 258:
4.6 proportion, not count, settles down

Sec. 4.2, p. 271 ff.
4.11 sample space
4.13 blood type
4.15 prob. assignments

equal likelihood and independence
4.12& 4.20, D&D
4.14 blood type again
4.32 GR...
 Read, discuss
Some situations where the probability
 is not "obvious" from a model :
p. 257, 4.1, 4.2, 4.3, 4.7

Optional
 More of the same:
4.10, 4.16, 4.17, 4.18

Blood type inheritance, 4.36-39

Making up points on the In-class exam:  You can earn back up to 50% of lost points on a problem by writing a new exam problem that tests the same things I was testing for, and doing the problem correctly.  Hand in the new problem(s), the solution(s), and your (old) exam.
For problem 10, the proofs, learn them, and do them orally on the board, with me asking at each stage why you did what you did, why it is "legal."  Please ask me for help beforehand on any of these.  Due on or before Mon. Oct 24
Add your data to the circulating sheets (3.75 grades, 3.73 Applet coin flips.   3.77 mean GPA)

3.4 conclusion
Note,   Sample size determines variability of sampling distribution: Population size doesn't matter p.238 (as long as pop. is at least "100" times the sample size (even at 10 times, you still aren't getting enough of the population. to have much effect.)  A spoonful of soup gives you just as good a sample, whether from a small pot or a caldroun.  A toothpickful of soup is just as bad...

I owe you:  Random number distributions have more variability than you might expect.

Pooled data:  phat Spinning penny (problem 4.1)
Results = part of Sampling distributions of the statistics which are used to estimate   the parameters p and mu.

Chapter 4: Probability

 Sec. 4.1
Random phenomenon= Chance mechanism. Any individual outcome is uncertain, BUT in a large number of repetitions, a regular distribution of outcomes occurs.
Variability happens.  In the long run it settles down.
 
Probability of any outcome of a random phenomenon:   proportion of times it would happen in a very long series of independent repetitions (trials) of the phenomenon: Long-term relative frequency.
(This is the "frequentist" interpretation. Mainstream)
   (independence:  outcome of one trial must not influence the outcome of any other.)
 http://www.whfreeman.com/ips/, Probability applet

Fuzzy word:  Random numbers  are each equally likely.  A random phenomenon  doesn't necessarily have outcomes that are equally likely--all that is needed is that their relative frequencies settle down to something.

Sec. 4.2, Probability Models

A Random phenomenon:  
Sample space S= Set of all possible outcomes.  (List or description, no overlaps )
  Event = outcome or set of outcomes (subset of S)
  Probability model: S, and a way of assigning a probability to each event.

Probability rules (axioms):  A an event in sample space S, P(A) is "the probability that  A occurs" p. 262
    1.  0 < P(A) < 1
    2. P(S) = 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) = P(Ac) = 1 - P(A)    Ac = "A complement"

ANY method of assigning probabilities to sets must result in a system that obeys these.
Note:  All the following obey Rules 1 thru 4:   All can be used as metaphors or models for probability models.
           Area   (If area of whole space is 1)  Venn diagrams of Axioms
          Volume (If volume of whole space is 1)
           Weight (If weight of whole space is 1)
           Proportion (count of items in set/ total count)

Equally likely probabilities:  If there are k possible outcomes, and each is equally likely, each has probability 1/k (Dice, coin, etc. Plenty of things are not equally likely.)

Pick one person from U.S. Pop. (Age 25 +)  Note how we 
can change a population distribution into a  probability mechanism by saying "choose an 
individual at random from the population. "  (problem 4.17--different pop.)
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      |

Flipping-coin-twice was built from a simpler phenomenon; flipping coin once: P(H) = .5, P(T) = .5
        \second
 first   H    T
    H | HH   HT 
    T | TH   TT    

Rule 5. Multiplication rule for Independent events.  If A and B are two independent events, the probability that both A and B occur  is the product of the probabilities of the two events.  P(A and B) = P(A)×P(B), if (and only if) A and B are independent.
   Independence (rule 5: Multiplication rule) is used two ways :
    (a) When building a complex probability model from simpler ones,  using the mult. rule to assign probabilities .
    (b) Checking if two events are independent in a known model, by seeing  if the mult. rule holds.

(a). Pick 2 people at random from U.S. pop.  (Pop. is so big that it's hardly changed by removing first. Independence OK)
   P(First has 4+ yrs college, and 2nd didn't graduate HS) = .280×.183 = .051
   P(First didn't graduate HS, and 2nd has 4+ yrs college) = .183×.280 = .051
   P(one didn't graduate HS, and the other has 4+ yrs college) = .051+.051= .102
 
(b)  Working Status vs. Sex.  Choose one person:  Is being a Woman independent  of being in the Labor Force?
P(Woman) = 500/1000 = .5, P(In L.F.) = 800/1000 = .8.  Product = .40 
P(Woman and In L.F.) = 350/1000 = .35  Not equal to .40, so not independent.  
  population of 1000
 Women Men Total
In Labor Force 350 450 800
Not in Labor Force 150 50 200
Total 500 500 1000

Statistics  question: if the 1000 is a sample from a larger population, is the difference from independence we see great enough to exclude the possibility that  Women/In L.F. are independent in the population; or is the difference just the result of sampling variability?

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