Math 300 , Spring 2004, Day 4, Monday, Feb 9 Hit reload to get most current versionAfter class

Move Classroom to 121(?) in long hall, permanently!.
HW from 1.3:
A) Handout on Hypergeometric distribution.
B)  Use Siegrist's Virtual Laboratories  (  http://www.math.uah.edu/stat/) to simulate the Hypergeometric distribution:
Bottom of main page, Applets.  Finite Sampling Models> Ball and Urn Experiment.. (The i button gives good info about the applets in general, but has the wrong labels for the Hypergeometric parameters.  Hover over the options for labels, or scroll down for text.)
Use 12 balls,  4 red, draw a sample of size 3. Single step a few times to see how it goes. Check your computation for the distribution in part A above. Reset.  Set Update 10, Stop 100.  Run.  See how close the bar graph of data matches (doesn't).  Run again and again (without resetting) until you like the match.  How many runs did you take? (scroll down the left data panel to see the last run)
Play with the pop. size, proportion of red, size of sample other values, look at the shape of the distribution in general.
NEW AFTER CLASS:  We found for hypergeometric,  n objects, k "defective",  a sample of r,
P(x defectives) = (kCx ×(n-k)C(r-x)) ÷ nCr.  In class we noted that this only works for x < r, x < k.  We also noted that when n=5, k=3, r=3, you could not have x =0 because there were not enough "good" objects.
Bb)  Find a general inequality involving x, with n, k, r, to reflect the possibility of not enough good objects.

 Ash, p. 19:
4 (like example 4)
5 (It's like the mosquitos.  They mean "exactly" 3 W and 2R)

3 (harder? I think it's easier with an ordered list of 3.  After you've chosen the first person, who CAN'T you choose?)
6 Hint: let the people pick the seats.
10 Hints:  1) Will the probability be the same whether i = 1, 2, or 3...?. .  2) I suggest doing it with i = 1, for 2 boys, 3 girls, then building up, 4 boys, 6 girls.  Look for a general approach.
11 Optional--read answers.  I don't want to put bad ideas in your head.

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"New" material: Sets--OR (union), AND (intersection), NOT (complement), and probabilities thereof
Ash 1-4, pp 20-27  (postpone P(AorBorC) pp21 on first reading)
Math 251 did a little of this: M&M 3rd ed: pp295-301 (not Independence), 346-350 for and, or (notConditional)
or M&M 4th ed: pp287-294 (not Independence), 346-343 for and, or  (texts are identical)

(Relative) Area is a good metaphor for probability.  Venn Diagrams help with counting too.
how to shade venn diagrams,    probabilities, diagrams for more than 3 sets (math research)

Probability RULES:  S is sample space, A event.  (M&M numbering)
1)  0< P(A) < 1,  any A in S
2)  P(S) = 1
3)  P(Ac) = 1- P(A)      (Therefore P(empty set, impossible event) = 0)
4)  If A and B are disjoint ((A and B) is empty), then P(Aor B) = P(A) + P(B)
4b) P(AorB) = P(A) + P(B) - P(A and B)     (4 is a special case of 4 b, but more "basic")
?    P (A and Not B) = P(A) - P(A and B).  (M&M fig. 4.19(ed.3), fig 4.18(ed.4))

Some Useful set rules:  (Ash's examples, p. 27)
"At least 3"   List numbers and circle your set, (bolded here)
0 1 2  3 4 5 6...  See what's left if you want the complement.  Why think if you can draw?
DeMorgan's Laws:
   (A or B)c = Ac and Bc       Not( Aor B) = Not A And Not B.
              I don't want cake or pie = I don't want cake and I don't want pie.
  (A and B)c = Ac or  Bc        Not (A and B) = Not A Or Not B:
             You can't have cake and pie = You can't have cake, or you can't have pie.

Distributive:  A and (B or C) = (A or B) and (A or C)
                     A or (B and C) = (A and B) or (A and C)  (not in texts but sometimes useful)

HW: on 1.4
C)  Show DeMorgan's laws true by shading a series of Venn diagrams.
From Moore&McCabe: 3rd ed: p. 359, 4.75, 4.76, 4.77, 4.78 or  4th ed: p. 353, 4.86,4.88 4.89 4.91 (2 pairs of problems that are the "same", numbers are different. (and numbers are different between editions)  Solutions to odd#s are in back of the book, 4th ed.sol's in the Math Clinic.
(Not for tonight, but next:  Ash, p.27:  3, 4, 5, 2, 8, 9, 11 a&b.   Don't feel bad if you have to look at the answers.)
 
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