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```Counting Principals
Ex. Eight pieces of paper are numbered 1 to 8 and placed
in a box. One piece of paper is drawn from the box,
its number is written down, and the piece of paper is
replaced in the box. A second piece of paper is drawn
from the box, and its number is written down. How
many different ways can a sum of 12 be obtained?
Ex. Eight pieces of paper are numbered 1 to 8 and placed
in a box. One piece of paper is drawn from the box,
its number is written down, and the piece of paper is
NOT replaced in the box. A second piece of paper is
drawn from the box, and its number is written down.
How many different ways can a sum of 12 be
obtained?
Ex. How many different pairs of letters from
the English alphabet are possible?
Ex. How many different phone numbers are
there in one area code? [Keep in mind
A permutation is the rearrangement of
elements.
Ex. How many permutation are there of the
letters A, B, C, D, E, and F?
The number of permutations of n elements is n!
Ex. How many distinguishable ways can the
letters in BANANA be written?
Ex. Eight horses are running in a race. How
many different ways are there for the
horses to take 1st, 2nd, and 3rd place?
The number of permutations of n elements
taken r at a time is
n!
nP
r
nr!
In the last example,
8
!
8

7

6

5

4

3

2

1
P



8

7

6
8
3
5
! 5

4

3

2

1
A combination is a list of elements where
order is not important.
{A, B, C} and {B, C, A}
are different permutations (where order
matters) but the same combination (where
order doesn't matter)
The number of combinations of n elements
taken r at a time is nCr
Ex. A standard poker hand consists of five
cards dealt from a deck of 52. How many
different poker hands are possible?
Ex. A 12-member swim team is being formed
from 10 girls and 15 boys. If the team must
consist of five girls and seven boys, how many
different teams are possible?
Practice Problems
Section 8.6
Problems 5, 7, 13, 19, 39, 43, 55
Probability
A sample space is a list of possible outcomes
Ex. Find the sample space if
a) one coin is tossed
b) two coins are tossed
The probability of an event is
n
u
m
b
e
r
f
a
v
o
r
a
b
l
e
P
e
v
e
n
t



n
u
m
b
e
r
p
o
s
s
i
b
l
e
Probabilities are always between
0 (impossible) and 1 (certain)
Ex. Two coins are tossed. What is the
possibility of both landing on heads?
Ex. A card is drawn from a deck, what is the
probability that it is an ace?
Ex. Two dice are tossed. What is the
probability that the total of the dice is 7?
52 Cards
Face Cards
♣
♠
♦
♥
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
9
9
9
9
10
10
10
10
J
J
J
J
Q
Q
Q
Q
K
K
K
K
A
A
A
A
Ex. What is the probability of winning a
lottery where a player chooses six
numbers (in any order) between 1 and 41?
P
A
o
r
B

P
A

P
B

P
A
a
n
d
B








Ex. When drawing a card from a standard
deck, what is the probability that the card
is either a heart or a face card?
♣
♠
♦
♥
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
9
9
9
9
10
10
10
10
J
J
J
J
Q
Q
Q
Q
K
K
K
K
A
A
A
A
Two events are independent if their occurrences don't
affect each other
If A and B are independent, then
P
A
a
n
d
B

P
A

P
B






Ex. A random number generator selects three integers
from 1 to 20. What is the probability that all three
numbers are less than or equal to 5?
Remember, a probability of 1 is a sure thing.
Sometimes it is easier to find the probability
of the complement of an event.
Ex. A manufacturer has determined that a machine
averages one faulty unit for every 1000 that it
produces. What is the probability that an order of 200
units will have one or more faulty units?
Practice Problems
Section 8.7
Problems 1, 11, 15, 17, 21, 45, 47, 48, 51
```
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