Transcript Chapter 3: Probability

Chapter 3
Probability
§ 3.4
Counting Principles
Fundamental Counting Principle
If one event can occur in m ways and a second event can
occur in n ways, the number of ways the two events can
occur in sequence is m· n. This rule can be extended
for any number of events occurring in a sequence.
Example:
A meal consists of a main dish, a side dish, and a dessert.
How many different meals can be selected if there are 4
main dishes, 2 side dishes and 5 desserts available?
# of main
# of side
dishes
dishes
4
2

There are 40 meals available.

# of
desserts
5
Larson & Farber, Elementary Statistics: Picturing the World, 3e
=
40
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Fundamental Counting Principle
Example:
Two coins are flipped. How many different outcomes are
there? List the sample space.
Start
1st Coin
Tossed
Heads
2 ways to flip the coin
Tails
2nd Coin
Tossed
Heads
Tails
Heads
Tails
2 ways to flip the coin
There are 2  2 = 4 different outcomes: {HH, HT, TH, TT}.
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Fundamental Counting Principle
Example:
The access code to a house's security system consists of 5
digits. Each digit can be 0 through 9. How many different
codes are available if
a.) each digit can be repeated?
b.) each digit can only be used once and not repeated?
a.) Because each digit can be repeated, there are 10
choices for each of the 5 digits.
10 · 10 · 10 · 10 · 10 = 100,000 codes
b.) Because each digit cannot be repeated, there are 10
choices for the first digit, 9 choices left for the second
digit, 8 for the third, 7 for the fourth and 6 for the fifth.
10 · 9 · 8 · 7 · 6 = 30,240 codes
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Permutations
A permutation is an ordered arrangement of objects. The n
umber of different permutations of n distinct objects is n!.
“n factorial”
n! = n · (n – 1)· (n – 2)· (n – 3)· …· 3· 2· 1
Example:
How many different surveys are required to cover all
possible question arrangements if there are 7 questions in
a survey?
7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040 surveys
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Permutation of n Objects Taken r at a Time
The number of permutations of n elements taken r at
a time is
n Pr 
# in the
group
n! .
(n  r)!
# taken from
the group
Example:
You are required to read 5 books from a list of 8. In how
many different orders can you do so?
n
Pr  8 P5 
8!  8! = 8  7  6  5  4  3  2  1  6720 ways
(8  5)! 3!
3  2 1
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Distinguishable Permutations
The number of distinguishable permutations of n objects,
where n1 are one type, n2 are another type, and so on is
n!
, where n1  n2  n3   nk  n.
n1 !  n2 !  n3 ! nk !
Example:
Jessie wants to plant 10 plants along the sidewalk in her
front yard. She has 3 rose bushes, 4 daffodils, and 3 lilies.
In how many distinguishable ways can the plants be
arranged?
10  9  8  7  6  5  4!
10!

3!4!3!
3!4!3!
 4,200 different ways to arrange the plants
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Combination of n Objects Taken r at a Time
A combination is a selection of r objects from a group of n
things when order does not matter. The number of
combinations of r objects selected from a group of n objects is
# in the
collection
nC r 
n!
.
(n  r)! r !
# taken from
the collection
Example:
You are required to read 5 books from a list of 8. In how
many different ways can you do so if the order doesn’t
matter?
8! = 8  7  6  5!
8C 5 =
3!5!
3!5!
= 56 combinations
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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Application of Counting Principles
Example:
In a state lottery, you must correctly select 6 numbers (in any order)
out of 44 to win the grand prize.
a.) How many ways can 6 numbers be chosen from the 44
numbers?
b.) If you purchase one lottery ticket, what is the
probability of winning the top prize?
a.)
44!
 7,059,052 combinations
C

44 6
6!38!
b.) There is only one winning ticket, therefore,
1
P (win) 
 0.00000014
7059052
Larson & Farber, Elementary Statistics: Picturing the World, 3e
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