Announcements
Finite Probability
Friday, September 9th
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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 I
Explain the multiplication principle intuitively
Use the multiplication principle to solve word problems
involving counting
Next Class: Sec. 5.4: Multiplication Principle II
Cherveny
Sept 9
Math 1004: Probability
Reading Questions
Focus questions from the reading:
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The figure on page 208 has 15 paths from A to C. How would
you explain to somebody that the answer is 15, perhaps using
the tree in figure two if you want.
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If you have four shirts, two hats, and three pairs of pants, how
many possible outfits do you have? How would you convince
somebody of this?
Cherveny
Sept 9
Math 1004: Probability
Multiplication Principle
Multiplication Principle If you have m ways to make a first
choice and n ways to make a second choice, then there are m · n
ways to make both choices together.
Proof.
Convince yourself by drawing a tree to illustrate the options.
Cherveny
Sept 9
Math 1004: Probability
Example
Example
Suppose that a man has 7 hats and 4 canes.
(a) How many ways can he choose a hat and a cane to go for a
walk?
Answer: 7 · 4 = 28 ways
(b) How many ways can he choose just a hat to wear today and a
different hat to wear tomorrow?
Answer: 7 · 6 = 42 ways
(c) How many ways can he choose a hat and cane for his walk
today and a hat and cane to go for his walk tomorrow?
Answer: 28 · 28 = 784 ways
Cherveny
Sept 9
Math 1004: Probability
Warning
What’s wrong? There are five toppings you can pick for a pizza, so
the number of ways you can get a pizza with two different toppings
is 5 · 4 = 20.
Answer: It’s wrong because a choice of topping A and then B
results in the same pizza as topping B and then topping A. When
using the multiplication principle, order matters.
This is OK: There are five toppings you can pick for a pizza, so the
number of ways you can tell the waiter the toppings for a two
topping pizza is 5 · 4 = 20.
Cherveny
Sept 9
Math 1004: Probability
Generalized Multiplication Principle
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 9
Math 1004: Probability
Example
Example
An quiz has five true/false questions and three multiple choice
questions (A to D options).
(a) If you didn’t study at all, how many ways can you fill in the
quiz?
Answer: 2 · 2 · 2 · 2 · 2 · 4 · 4 · 4 = 25 · 43 = 2, 048 ways
(b) How many ways can you fill in the quiz if you alternate your
answers to the true/false and don’t use the same multiple
choice answer twice?
Answer: 2 · 1 · 1 · 1 · 1 · 4 · 3 · 2 = 48 ways
(c) How many ways can you fill in the quiz if you only use false
once and you use all the same letter for the multiple choice?
Answer: 5 · 4 = 20 ways
Cherveny
Sept 9
Math 1004: Probability
Practice
1. Including nonsense words like “xqpi”,
(a) How many four letter words are there?
(b) How many four letter words don’t use a letter multiple times?
(c) How many four letter words are palindromes?
2. There are four men and four women.
(a) How many ways can they all stand in line against a wall?
(b) How many ways can five of them stand against the wall for a
group picture and one of them take the photo?
(c) How many ways can they all stand against the wall if they
alternate gender?
(d) Suppose they are all married (man and woman). How many
ways can they all stand in line for the photo if married couples
always stand together?
3. How many ways can a Venn diagram with two circles be
shaded? What about a Venn diagram with three circles?
Cherveny
Sept 9
Math 1004: Probability
Practice Answers
1. (a) 264 = 456, 976
(b) 26 · 25 · 24 · 23 = 358, 800
(c) 26 · 26 · 1 · 1 = 676
2. (a) 8! = 40, 320
(b) 8 · 7 · 6 · 5 · 4 · 3 = 20, 160
(c) 8 · 4 · 3 · 3 · 2 · 2 · 1 · 1 = 1, 152
(d) 4 · 2 · 3 · 2 · 2 · 2 · 1 · 2 = 384
3. There are 24 = 16 ways to shade a Venn diagram with two
circles? There are 28 = 256 ways to shade a Venn diagram
with three circles.
Cherveny
Sept 9
Math 1004: Probability