Announcements
Finite Probability
Monday, September 12th
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MyMathLab 2 is due Wednesday Sept 14
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Problem Set 2 is due Friday Sept 16
Today: Sec. 5.4: Multiplication Principle II
Explain the multiplication principle intuitively
Use the multiplication principle to solve word problems
involving counting
Next Class: Sec. 5.5: Combinations and Permutations I
Cherveny
Sept 12
Math 1004: Probability
Reading Questions
Focus questions from the reading:
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Consider again the maze on page 208. If you want to go from
A to B to C to B to A, how many ways can you do it? What
if you don’t want to use the same path twice?
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Suppose you have a bin of four different types of fruit and you
want to pick three of them for a picnic. How many ways can
you do it? What do you say to your friend that says the
answer is 4 · 4 · 4 = 64? What if they answer 4 · 3 · 2? What’s
the clearest way you can explain your answer?
Cherveny
Sept 12
Math 1004: Probability
Generalized Multiplication Principle
Last time:
Generalized Multiplication Principle If there are many k choices
to be made, and the number of options for each choice is
n1 , n2 , . . . , nk , then there are a total of n1 · n2 · · · · · nk ways to
make all the choices together.
Cherveny
Sept 12
Math 1004: Probability
Shading Venn Diagrams
Last practice question last lecture:
Example
How many ways can you shade a Venn diagram for 2 sets?
What about a Venn diagram for 3 sets?
Answer: Shading is determined by how each basic region is
shaded. A Venn diagram for 2 sets has 4 basic regions, and we
have two choices for each basic region (shaded or unshaded).
−→ A 2 set Venn diagram has 24 possible shadings.
Likewise, a 3 set Venn diagram has 8 basic regions and so there are
28 possible shadings.
Cherveny
Sept 12
Math 1004: Probability
Zip Codes
Example
Zip codes are read by machines to determine what region of the
US to route a letter. These machines need to be able to read zip
codes upside down.
We say a five-digit zip code is “detour prone” if it looks like a valid
and different zip code when read upside down. For instance, 68901
is detour prone, but 10801 is not. How many possible zip codes are
there if detour prone zip codes are not used?
Answer: 105 − (55 − 5 · 5 · 3 · 1 · 1) = 96950.
Trivia: There are actually about 43,000 zip codes in use.
Cherveny
Sept 12
Math 1004: Probability
Practice
1. You have 3 math books, 5 biographies, and 6 novels. How
many ways can the books be arranged on your shelf if..
(a)
(b)
(c)
(d)
There are no restrictions on arrangement?
All the books of each type must be next to each other?
There is at least one book not next to another book of its type?
All the biographies must be together and all the novels must be
together but no two math books are next to each other?
2. Two 10-member basketball teams play a game. Each of the
members of the winning team shakes hands once with each
member of both teams. How many handshakes take place?
3. 10 knights in King Arthur’s court need to sit at a round table.
How many seating arrangements are there if...
(a) There are no seating restrictions?
(b) Sirs C, D, and E must all be seated together?
(c) Sir A and Sir B are feuding and cannot be seated next to each
other?
Cherveny
Sept 12
Math 1004: Probability
Practice Answers:
1. (a) 14! = 87178291200
(b) 3!5!6!3! = 3110400
(c) 14! − 3!5!6!3! = 87175180800
(d) 5!6!3 · 2 · 1 · 1 · 1 = 518400
2. 102 + (10 · 9)/2 = 145 hand shakes
3. (a) 9! = 362880
(b) 3!7! = 30240
(c) 9! − 2 · 8! = 282240
Cherveny
Sept 12
Math 1004: Probability