Counting and Probability

1
Counting and Probability

• By Jeffrey Bivin
• Lake Zurich High School
• jeff.bivin_at_lz95.org

Last Updated April 20, 2009
2
Fundamental Counting Principal
How many different meals can be made if 2 main
courses, 3 vegetables, and 2 desserts are
available?
Lets choose a main course
2
M1 M2
Now choose a vegetable
3
x
V1 V2 V3 V1
V2 V3
Finally choose A dessert
2
x
D1 D2 D1 D2 D1 D2 D1 D2 D1
D2 D1 D2
12
1 2 3 4 5 6 7 8
9 10 11 12
3
Linear Permutations
A club has 30 members and must select a
president, vice president, secretary, and
treasurer. How many different sets of officers
are possible?
30
29
28
27



president
vice-president
secretary
treasurer
657,720
4
Linear Permutations
Alternative
A club has 30 members and must select a
president, vice president, secretary, and
treasurer. How many different sets of officers
are possible?
1
2
4
3
30
29
28
27



Try with your calculator
president
vice-president
secretary
treasurer
657,720
30P4
5
Permutation Formula
657,720
6
Linear Permutations
There are 25 students in a classroom with 25
seats in the room, how many different seating
charts are possible?
25
24
23
22
21




. . .
seat 1
seat 2
seat 3
seat 4
seat 5
1.5511 x 1025
25!
7
Linear Permutations
Alternative
There are 25 students in a classroom with 25
seats in the room, how many different seating
charts are possible?
25
24
23
22
21




. . .
seat 1
seat 2
seat 3
seat 4
seat 5
1.5511 x 1025
25!
25P25
8
Permutation Formula
1.5511 x 1025
9
More Permutations
There are 5 people sitting at a round table, how
many different seating arrangements are possible?
straight line
Divide by 5
A B C D E

E A B C D

D E A B C

C D E A B

B C D E A

10
More Permutations
REVIEW
There are 5 people sitting at a round table, how
many different seating arrangements are possible?
straight line
When circular, divide by the number of items in
the circle
A B C D E

E A B C D

D E A B C

Treat all permutations as if linear
Now consider the circular issue
C D E A B

B C D E A

11
More Permutations
There are 9 people sitting around a campfire, how
many different seating arrangements are possible?
straight line
Yes, divide by 9
Treat all permutations as if linear
Is it circular?
A B C D E F G H I
12
More Permutations
There are 5 people sitting at a round table with
a captain chair, how many different seating
arrangements are possible?
straight line
NOT CIRCULAR
A B C D E

E A B C D

D E A B C

NOTE
C D E A B

B C D E A

Each table has someone different in the captian
chair!
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More Permutations
How many ways can you arrange 3 keys on a key
ring?
straight line
Yes, divide by 3
Treat all permutations as if linear
Is it circular?
A B C
Now, try it. . .
PROBLEMTurning it over results in the same
outcome.
So, we must divide by 2.
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More Permutations
How many ways can you arrange the letters MATH ?
How many ways can you arrange the letters ABCDEF
?
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Permutations with Repetition
How many ways can you arrange the letters AAAB?
Divide by 3!
Lets look at the possibilities
If a permutation has repeated items, we divide by
the number of ways of arranging the repeated
items (as if they were different).
AAAB
AABA
Are there any others?
What is the problem?
ABAA
BAAA
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How many ways can you arrange 5 red, 7 blue and 8
white flags on the tack strip across the front of
the classroom?
If all were different, how may ways could we
arrange 20 items?
There are 5 repeated red flags ? Divide by 5!
There are 7 repeated blue flags ? Divide by 7!
There are 8 repeated white flags ? Divide by 8!
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How many ways can you arrange the letters
AABBCCCCDEFGGGGGG ?
If all were different, how may ways could we
arrange 17 items?
There are 2 repeated As ? Divide by 2!
There are 2 repeated Bs ? Divide by 2!
There are 4 repeated Cs ? Divide by 4!
There are 6 repeated Gs ? Divide by 6!
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Permutations
ORDER
Multiply the possibilities
or
Use the nPr formula (if no replacement)
Divide by the number of items in the circle
Divide by 2
Divide by the factorial of the number of each
duplicated item
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How many ways can you put 5 red and 7 brown beads
on a necklace?
How may ways could we arrange 12 items in a
straight line?
Is it circular? Yes ? divide by 12
Can it be turned over? Yes ? divide by 2
33
Are there repeated items? Yes ? divide by
5! and 7!
20
How many ways can you arrange 5 red and 7 brown
beads on a necklace that has a clasp?
How may ways could we arrange 12 items in a
straight line?
Is it circular? N0 ? the clasp makes it
linear
Can it be turned over? Yes ? divide by 2
396
Are there repeated items? Yes ? divide by
5! and 7!
21
How different license plates can have 2 letters
followed by 3 digits (no repeats)?
A straight line?
Is it circular? No
468,000
Can it be turned over? No
Are there repeated items? No
22
How different license plates can have 2 letters
followed by 3 digits with repeats?
A straight line?
Is it circular? No
676,000
Can it be turned over? No
Are there repeated items? Yes, but because
we are using multiplication and not factorials,
we do not need to divide by anything.
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Solve for n nP4 40 n-1P2
8
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Combinations
NO order NO replacement
Use the nCr formula
25
Combinations
An organization has 30 members and must select a
committee of 4 people to plan an upcoming
function. How many different committees are
possible?
27,405
26
Combinations
A plane contains 12 points, no three of which are
co-linear. How many different triangles can be
formed?
220
27
Combinations
An jar contains 20 marbles 5 red, 6 white and
9 blue. If three are selected at random, how
many ways can you select 3 blue marbles?
84
28
Combinations
An jar contains 20 marbles 5 red, 6 white and
9 blue. If three are selected at random, how
many ways can you select 3 red marbles?
10
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The OR factor.
An jar contains 20 marbles 5 red, 6 white and
9 blue. If three are selected at random, how
many ways can you select 3 blue marbles or 3 red
marbles?
OR ? ADD
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The AND factor.
An jar contains 20 marbles 5 red, 6 white and
9 blue. If three are selected at random, how
many ways can you select 2 blue marbles and 1 red
marble?
AND ? MULTIPLY
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At least
An jar contains 20 marbles 5 red, 6 white and
9 blue. If five marbles are selected at random,
how many ways can you select at least 3 blue
marbles?
3 or 4 or 5 blue
3B2NB or 4B1NB or 5B
32
At most
An jar contains 20 marbles 5 red, 6 white and
9 blue. If five marbles are selected at random,
how many ways can you select at most 1 red
marbles?
0 or 1 red
0R5Nr or 1R4NR
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Evaluate each of the following
34
SO..
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PROBABILITY

• Definition

number of success
The ratio ?
total number of outcomes
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Probability
A coin is tossed, what is the probability that
you will obtain a heads?
Look at the sample space/possible outcomes
H , T
number of success
1
Pr(H)

2
total number of outcomes
37
Probability
A die is tossed, what is the probability that you
will obtain a number greater than 4?
Look at the sample space/possible outcomes
1 , 2 , 3 , 4 , 5 , 6
number of success
2
1
Pr(gt4)


6
3
total number of outcomes
38
Probability Success Failure
A die is tossed, what is the probability that you
will obtain a number greater than 4?
number of success
2
1
Pr(gt4)


6
3
total number of outcomes
What is the probability that you fail to obtain a
number greater than 4?
number of failures
4
2
Pr(gt4)


6
3
total number of outcomes
1
Pr(success) Pr(failure) 1
TOTAL
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Probability
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that all three are red?
5C3
number of success
Pr(3R)

13C3
total number of outcomes
have
want
5 red 8 blue

3 red
Total 13 ? 3
40
Probability
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that all three are blue?
8C3
number of success
Pr(3B)

13C3
total number of outcomes
have
want
5 red 8 blue

3 blue
Total 13 ? 3
41
Probability and
multiply
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that one is red and two are blue?
5C1 ? 8C2
number of success
Pr(1R2B)

13C3
total number of outcomes
have
want
5 red 8 blue

1 red
and
2 blue
Total 13 ? 3
42
A jar contains 5 red, 8 blue and 7 white marbles.
If 3 marbles are selected at random, what is the
probability that one of each color is selected?
1 red, 1 blue, 1 white
5C1?8C1?7C1
of success
Pr(1R,1B,1W)

20C3
total of outcomes
have
want
5 red 8 blue 7 white

and
1 red
and
1 blue
1 white
Total 20 ? 3
43
A jar contains 7 red, 5 blue and 3 white marbles.
If 4 marbles are selected at random, what is the
probability that 2 red and 2 white marbles are
selected?
7C2 ? 3C2
of success
Pr(2R,2W)

15C4
total of outcomes
have
want
7 red 5 blue 3 white

2 red
and
2 white
Total 15 ? 4
44
Five cards are dealt from a standard deck of
cards. What is the probability that 3 hearts and
2 clubs are obtained?
13C3 ? 13C2
of success
Pr(3H,2C)

52C5
total of outcomes
have
want
13 diamonds 13 hearts 13 clubs 13 spades

and
3 hearts
2 clubs
Total 52 ? 5
45
Probability or
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that all three are red or all three
are blue?
5C3 8C3
of success
Pr(3R or 3B)

13C3
total of outcomes
have
want
want
5 red 8 blue


3 red

OR
3 blue
Total 13 ? 3
46
A jar contains 5 red and 8 blue marbles and 7
yellow marbles. If 3 marbles are selected at
random, what is the probability that all three
are the same color?
3 red or 3 blue or 3 yellow ?
5C3 8C3 7C3
of success
Pr(3R or 3B or 3Y)

20C3
total of outcomes
have
want
want
want

5 red 8 blue 7 yellow



3 red
OR
OR
3 blue
3 yellow
Total 20 ? 3
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Probability or with overlap
If two cards are selected from a standard deck of
cards, what is the probability that both are red
or both are kings?
Pr(2R or 2K) Pr(2R) Pr(2K) Pr(2RK)
26C2 4C2 2C2
of success

52C2
total of outcomes
have
want
want

overlap
26 red 26 black

2 red

2 red kings
OR
4 kings 48 other

2 kings
Total 52 ? 2
48
Probability and with or
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that two are red and one is blue or
that one is red and two are blue?
5C2? 8C1 5C1 ? 8C2
of success
Pr(2R1B or 1R2B)

13C3
total of outcomes
have
want
want
5 red 8 blue


2 red
1 red

OR
and
and
1 blue
2 blue
Total 13 ? 3
49
Probability at least
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that at least two red marbles are
selected?
2 red and 1 blue or 3 red
2 red or 3 red
5C2? 8C1 5C3
of success
Pr(at least 2Red)

Pr(2R1B or 3R)
13C3
total of outcomes
Wait, we need 3 marbles!
have
want
want
5 red 8 blue


2 red
3 red

OR
and
1 blue
Total 13 ? 3
50
Probability at least
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that at least one red marble is
selected?
5C1? 8C2 5C2 ? 8C1 5C3
Pr(at least 1Red)
Pr(1R2B or 2R1B or 3R)
13C3
Remember, we need 3 marbles!
have
want
want
want

5 red 8 blue



1 red
2 red
3 red
OR
OR
and
and
2 blue
1 blue
Total13 ? 3
51
Probability at least
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that NO red marbles are selected?
8C3
Pr(0R3B)
13C3
In the previous example we found
have
want
5 red 8 blue

3 blue
Pr(success) Pr(failure) 1
Total13 ? 3
52
Probability at least
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that at least one red marble is
selected?
Pr(success) Pr(failure) 1
Pr(success) 1 - Pr(failure)
Pr(gt1 red) 1 Pr( 0 red )
Pr(3 blue)
53
Probability at least
A jar contains 8 red and 9 blue marbles. If 7
marbles are selected at random, what is the
probability that at least one red marbles is
selected?
FASTEST
Pr(at least 1Red)
success
Pr(1R6B or 2R5B or 3R4B or 4R3B or 5R2B or 6R1B
or 7R)
Pr( 0R7B )
Pr(0Red)
failure
Pr(at least 1Red) 1 - Pr(0R7B)
54
Probability at least
A jar contains 8 red, 9 blue and 3 white marbles.
If 7 marbles are selected at random, what is the
probability that at least three red marbles are
selected?
FASTEST
Pr(gt 3Red) ? Pr(3-7 red)
success
Pr(lt 3Red) ? Pr(0-2 red)
failure
1 - Pr(0R7NR or 1R6NR or 2R5NR)
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Probability with replacement
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that one red followed by two blue
marbles are selected if each marble is replaced
after each selection?
Note In this example an order is specified
Must use fractions!
R B B
56
Probability with replacement
A jar contains 5 red and 8 blue marbles. If 3
marbles are selected at random, what is the
probability that one red and two blue marbles
are selected if each marble is replaced after
each selection?
Problem Fractions imply order!
Must use fractions!
R B B
Must account of any order!