Transcript STAT 113 - Purdue University

Chapter 4.1:
Combinatorics and Basic Counting Rules
Chris Morgan, MATH G160
[email protected]
January 25, 2012
Lecture 7
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Combinatorics
Many events have far too many outcomes to list all of them:
- How many possible outcomes are there in flipping a coin
six times? (64)
- How many possible ways can I get a Jimmy John’s
sandwich made? (a lot)
- How many ways can I rearrange all of you in your desks?
(even more ways)
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Combinatorics
• Listing all possible situations is unpractical, and usually we don’t
care about each individual outcome (only care about total
outcomes)
• Combinatorics is the study of “counting rules” so we can count
more quickly
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Example (I)
Cassie owns 2 different pairs of shoes, 4 different shirts, and 3
different pairs of pants. How many different outfits can she wear?
2 * 4 * 3 = 24
shoes
shirts
pants
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Example (I)
Shoes
Shirts
Pants
Shoes
Shirts
Pants
Shirts
Pants
Shirts
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Example (I)
Shoes
Shirts
Pants
Shoes
Shirts
Pants
Shirts
Pants
Shirts
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Basic Counting Rule
Suppose that r actions (choices, experiments) are to be performed
in a definite order, further suppose that there are m1 possibilities
for the first action, m2 possibilities for the second action, etc,
then there are m1 × m2 × . . . × mr total possibilities altogether.
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Basic Counting Rule
To start, let’s demonstrate BCR by tossing a coin twice:
How many possible ways are there to toss the coin?
How many possible ways are there to toss the coin?
- m1 = 2 (Heads or Tails)
- m2 = 2 (Heads or Tails)
2
* 2
=
4
H
T
H
T
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Basic Counting Rule
• A phone number consists of 10 numbers. The prefix for
Indianapolis is 317, how many phone numbers are possible?
10 digits, but the first three are set: 3 1 7 _ _ _ _ _ _ _
1 * 1 * 1 * 10 * 10 * 10 * 10 * 10 * 10 * 10
= 107 = 10,000,000
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Basic Counting Rule
What is the probability that a phone number
contains the numbers 0-6 in any order?
_
_
_
_
_
_
_ = ??
7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040
P(0-6) = 5, 040 / 10,000,000
= 0.000504
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Example (II)
Assuming Mary has 6 pairs of shoes, 10 different tops, 8 different
bottoms and 4 different outwears, then how many combinations can
she have for outfit?
Mary is having a job interview and she wants to decide what to wear. If
there are 2 pairs of shoes, 3 tops, 2 bottoms and 2 outwears that are
appropriate for an interview and she randomly picks what to wear
for the interview among all she has, what is the probability that she
wears an interview-appropriate outfit?
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Example (III)
Mary bought a lock for her new bike since her last one was stolen.
There are 4 slots numbered 0 to 9, how many possible combinations
are there?
• If the combination only includes even numbers
• If the first number can not be 0 and all four numbers must all
be different
• If the combination must have at least one 4 or at least one 5
• What are the probabilities for those specified combination?
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Example (IV)
At a Subway, you have to decide what you want to put in your
sandwich, the choices you have are:
• Four types of bread
• Five types of cheese
• Six types of veggies
• Seven types of meat
Assuming you can only choose one from each of the above categories,
how many total possible combinations could we get?
If I don’t like white bread, only like Swiss cheese, don’t like onion, and
am allergic to seafood and chicken, what is the chance that I get a
sandwich that I actually like?
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Example (V)
An assiduous student named Sam finds himself hungry at 2 a.m. on a Tuesday.
This unremitting undergraduate has become conscious of a considerable
craving for Connie’s pizza. Alas, he is nowhere near Chicago to fulfill such a
phenomenal food fantasy. He has to settle for 1 of 5 pizza places that still
permits pie purchases. Each restaurant has 3 choices for crust type: thin
crust, regular, and deep dish. Additionally, a customer is allowed to have at
most 1 meat out of the 4 total choices and at most 1 vegetable out of the 5
for his toppings. How many possible pizzas could the famished freshman
feast on?
Ceteris paribus, how many possible pizzas are there if the place permits at
most 2 meat choices?
Ceteris paribus, how many pizzas are possible if the restaurant allows at most 2
meat choices and any combination of vegetables?
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Example (VI)
A young man wants to plan a nice date for his girlfriend. He has the option of
going to Chicago, Indianapolis, or staying Lafayette. If he chooses Chicago,
he has 10 choices for a play and 100 choices for a restaurant. If he opts of
Indianapolis, he has 5 choices for a play and 50 choices for a restaurant. If
he remains in Lafayette, he only has 2 options for a play 20 choices for a
restaurant.
How many options does this gentleman have for a romantic evening out?
What about if we added the option of Fort Wayne and this particular city
boasted 8 plays and 75 restaurants?
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Example (VII)
N‐ is population size (sampling) and n‐ is sample size
Sampling with replacement:
•
•
How many possible ordered samples of size 3 with replacement from a
population of size 4?
In general, how many possible ordered samples of size n with replacement
from a population of size N.
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