Read Ash 1-5: Continuous uniform distributions:
HW Ash p. 34, 1, 2, 6, 7.
Read #3 and its answer.
A) Suppose I have a cardboard
circle whose diameter is half the length of one of the square
tiles on the classroom floor.
The experiment is to toss the circle on the floor. What is the
probability that the circle lands wholly within a square (not touching
the sides)?
Assume that any place on the floor is as likely as any other
place. Hint: Think about where the center of the circle
lands, do the geometry using that.
Ash Ch.2.We'll use Moore&McCabe 5th ed. section 4.5 with
Ash 2.1(conditional prob. and independence) &2.4 (sequences and
Bayes'),
then
review section M&M 5.1 with Ash 2.2 (Binomial, multinomial)
Most of Ash's examples stay with the "equally likely" model of
probability, but often the work extends to the other models as well.
Conditional probability and chain rules:
3 presentations: Venn diagram, two-way table
(both good for two "things"), tree (especially good for causal
or decision sequence; can work for sequence of 2, 3, or more
things).
Reading (easiest mentally): Ash 2-1, read to bottom of
p. 39 (def. of cond. prob),
Moore&McCabe 5th ed., pp.315-321
(conditional probability,
chain ("multiplication") rule, tree diagrams)
+ pp322-3 ("Independence Again")
Ash, Chain rules for AND, pp. 39-41 (including
independence), then "Theorem of Total
probability" pp.57-8 (stop before Bayes' for now)
Handout: "Decision
Analysis" (from M&M 4th ed, pp. 350-352)
HW
on conditional probability. Draw trees whenever possible
or appropriate; two-way table or Venn diagram if that is most
appropriate.
(When reasonable, try to represent the problem 2 or more ways.
)
Moore&McCabe 5th, p. 323
ff. 4.95, 4.96, 4.97
On separate paper (and keep) 4.104, 105.
(I'll assign 4.106, 107 with Bayes' theorem.)
Handout
(Back of "Decision Analysis" ) Old M&M 4.95 (this
is an example of the Geometric distribution,
which we'll meet later.)
, 4.96
Ash sec. 2-1, p. 44
1, 2, 3,
4 (this is a workout of all our techniques at once)
6 Draw a picture, use geometry.
Sec 2-4, p. 61 (trees)
9
1, 2,
7
Separate paper, keep: 3a (3b will come with Bayes' th.)
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Notes:Ash 1-5: Continuous uniform distributions:
When any point in an interval or in an area is as likely as
any
other ("uniform" distribution),
then probabilities can be calculated directly from lengths, or from
areas.
Bulletin board: I throw a dart at it. Suppose I am bad enough
at throwing darts that every point on the bulletin board is as
likely to be hit as any other point. What is the probability that
my dart lands in the circle? P = (Area of circle)/(area of whole
square).
2 independent Spinners: edge labeled 0.0 to1.0. (Choose 2 numbers
at random between 0 and 1.) Results are X and Y. What is
P(X>Y)? Intuitively 1/2: What region of the x/y square
corresponds to x>y? (Remember x=y has prob. 0)
(Note--from review, Ash p. 29: to tell which side of a boundary line
satisfies
an inequality, it is enough to test one point.)
I'll discuss Buffon's needle (pp. 33-4) maybe
Monday.
- - - - - - - - - - - - - - - - - - -
Recall: There are 3 ways of connecting
probability theory as mathematics
with the real world.
Classical = equally likely = "model based"
Relative frequency = "frequentist"
Subjective = "Bayesian" = "personal"
Conditional probability and chain rules: (Ash 2-1, 2-4
to p. 58)
P(A | B) = P (A and B)
P( B)
Prob of A given B = Prob of (A and B) divided by Prob of
B
A Given B: B is the whole thing now!
Throw away the part of A not in
B:
P(A and B)
Divide by P(B): standardizes (A and
B) as a proportion of the "whole" B.
Look out the window, see Snow, but can't tell if the wind is
blowing. Know the probabilities for this season. Prob(Wind|
Snow) = .3/.4 (see picture)
Another day, hear the Wind but the sun's not up; can't see if it's
snowing.
P(Snow| Wind) = P(W&S)/P(W) = .3/.5. Same
numerator, different denominator!
Another day, see it's Not Snowing. P(Wind| Not Snow) = P(W&Sc)/P(Sc),
have some figuring to do...
3 presentations: Venn diagram
(this privileges sets over complements)
Two-way table (Table expands to
more possibilities (e.g. wind absent, light, heavy), doesn't privilege
anything),
| S | Sc | total | Bold numbers are basic numbers from
Venn diagram. All other numbers follow. P(W|S) = W &S cell divided by S column total. P(Wind| Not Snow) = .2/.6 |
|
| W | .3 | .2 | .5 | |
| Wc | .1 | .4 | .5 | |
| total | .4 | .6 | 1.0 |
There is often a confusion in textbooks between the conditional
probability
as:
(1)--Something calculated from a known Sample Space or
probability
model: partial knowledge is obtained (B is true, has
occurred, is
perceived) which changes probabilities assigned to other events.
(Have Venn Diagram or Table info.)
(2)--The formula used to construct a probability model when
the model has a natural causal or decision chain, the probabilities at
each step are known given the previous step, and the task is to
construct
probabilities for events at the end of the chain especially (and maybe
for the intermediate events also.).
Philosophically, these are different. However, they are consistent. If you use chaining or a tree to construct a model, and then calculate the conditional probabilities for pieces in the tree from your total sample space, they will come out with the same values as the ones you used in constructing the tree. (Whew!) I think it's helpful to keep track of whether you're doing (1) or (2) at any given moment.
Ash Example 3, p. 40: Chain rule (Multiplication rule) represented
as tree.
(Aside. You could answer this with perms or combs also.)
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