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Jeff Bivin -- LZHS
Counting and Probability
By: Jeffrey BivinLake Zurich High School
jeff.bivin@lz95.org
Last Updated: April 16, 2008
Jeff Bivin -- LZHS
Fundamental Counting PrincipalHow many different meals can be made if 2 main courses, 3 vegetables, and 2 desserts are available?
M1 M2
V1 V2 V3 V1 V2 V3
D1 D2 D1 D2 D1 D2 D1 D2 D1 D2 D1 D2
1 2 3 4 5 6 7 8 9 10 11 12
Let’s choose a
main course
Now choose a
vegetable
Finally choose
A dessert
Jeff Bivin -- LZHS
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?
president vice-president secretary treasurer
Jeff Bivin -- LZHS
A club has 30 members and must select a president, vice president, secretary, and treasurer. How many different sets of officers are possible?
president vice-president secretary treasurer
30P4
Linear Permutations
Jeff Bivin -- LZHS
Permutation Formula
)!(
!
rn
nPrn
!26
!30
)!430(
!30430
P
27282930!26
!2627282930
Jeff Bivin -- LZHS
Linear Permutations
There are 25 students in a classroom with 25 seats in the room, how many different seating charts are possible?
seat 1 seat 2 seat 3 seat 4 seat 5
1.5511 x 1025
Jeff Bivin -- LZHS
Linear Permutations
There are 25 students in a classroom with 25 seats in the room, how many different seating charts are possible?
seat 1 seat 2 seat 3 seat 4 seat 5
25P25
1.5511 x 1025
Jeff Bivin -- LZHS
Permutation Formula
)!(
!
rn
nPrn
!0
!25
)!2525(
!252525
P
!251
!25 1.5511 x 1025
Jeff Bivin -- LZHS
More PermutationsThere are 5 people sitting at a round table, how many different seating arrangements are possible?
245
120
5
!5
straight line
Divide by 5
Jeff Bivin -- LZHS
More PermutationsThere are 5 people sitting at a round table, how many different seating arrangements are possible?
245
120
5
!5
straight line
Treat all permutations as if linear
Now consider the circular issue
When circular, divide by the number of items in the circle
Jeff Bivin -- LZHS
More PermutationsThere are 9 people sitting around a campfire, how many different seating arrangements are possible?
403209
362880
9
!9
straight line
Treat all permutations as if linear
Is it circular?
Yes, divide by 9
Jeff Bivin -- LZHS
There are 5 people sitting at a round table with a captain chair, how many different seating arrangements are possible?
More Permutations
120!5
straight line
NOTE:
Jeff Bivin -- LZHS
More PermutationsHow many ways can you arrange 3 keys on a key ring?
12
236
3!3
straight line
Treat all permutations as if linearIs it circular?
Now, try it. . .PROBLEM:Turning it over results in the same outcome.
Yes, divide by 3
So, we must divide by 2.
Jeff Bivin -- LZHS
More PermutationsHow many ways can you arrange the letters MATH ?
24!4
How many ways can you arrange the letters ABCDEF ?
720!6
Jeff Bivin -- LZHS
Permutations with RepetitionHow many ways can you arrange the letters AAAB?
46
24
!3
24!4
Let’s look at the possibilities:
AAABAABAABAABAAA
Are there any others?What is the problem?
If a permutation has repeated items, we divide by the number of ways of arranging the repeated items (as if they were different).
Divide by 3!
Jeff Bivin -- LZHS
How many ways can you arrange 5 red, 7 blue and 8 white flags on the tack strip across the front of the classroom?
240,768,99!8!7!5
!20
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!
Jeff Bivin -- LZHS
How many ways can you arrange the letters AABBCCCCDEFGGGGGG ?
800,940,145,5!6!4!2!2
!17
If all were different, how may ways could we arrange
17 items?
There are 2 repeated A’s Divide by 2!
There are 2 repeated B’s Divide by 2!
There are 4 repeated C’s Divide by 4!
There are 6 repeated G’s Divide by 6!
Jeff Bivin -- LZHS
Permutations ORDER
Multiply the possibilities
Divide by the numberof items in the circle
Divide by 2
Divide by the factorial of thenumber of each duplicated item
Assume the itemsare in a straight line! Use the nPr formula
(if no replacement)
or
Are the items in a circle??
Can the itembe turned over??
Are there duplicateitems in your
arrangement??
Jeff Bivin -- LZHS
How many ways can you put 5 red and 7 brown beads on a necklace?
!7!5212
!12
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
Are there repeated items? Yes divide by 5! and 7!
33
Jeff Bivin -- LZHS
How many ways can you arrange 5 red and 7 brown beads on a necklace that has a clasp?
!7!52
!12
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
Are there repeated items? Yes divide by 5! and 7!
396
Jeff Bivin -- LZHS
How different license plates can have 2 letters followed by 3 digits (no repeats)?
A straight line?
Is it circular? No
Can it be turned over? No
Are there repeated items? No
468,000
26 ∙ 25 ∙ 10 ∙ 9 ∙ 8lette
rletter number number number
Jeff Bivin -- LZHS
How different license plates can have 2 letters followed by 3 digits with repeats?
A straight line?
Is it circular? No
Can it be turned over? No
Are there repeated items? Yes, but because we are using multiplication andnot factorials, we do not need to divide by anything.
676,000
26 ∙ 26 ∙ 10 ∙ 10 ∙ 10lette
rletter number number number
Jeff Bivin -- LZHS
Combinations NO orderNO replacement
Use the
nCr formula
Jeff Bivin -- LZHS
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?
!)!(
!
rrn
nCrn
!4!26
!30
!4)!430(
!30430
C
Jeff Bivin -- LZHS
!3!9
!12
!3)!312(
!12312
C
Combinations
!)!(
!
rrn
nCrn
A plane contains 12 points, no three of which are co-linear. How many different triangles can be formed?
Jeff Bivin -- LZHS
!3!6
!9
!3)!39(
!939
C
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?
Combinations
!)!(
!
rrn
nCrn
Jeff Bivin -- LZHS
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?
!3!2
!5
!3)!35(
!535
C
Combinations
!)!(
!
rrn
nCrn
Jeff Bivin -- LZHS
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?
The OR factor.
OR ADD
!3)!35(
!5
!3)!39(
!93539
CC
941084
Jeff Bivin -- LZHS
The OR factor.
10 84 94 5 red
6 white9 blue
3 redOR
3 blue
have want
5 3 9 3
5! 9!
(5 3)! 3! (9 3)! 3!C C
want
OR ADD
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 or 3 blue marbles?
Jeff Bivin -- LZHS
The OR factor.
5 70 75 5 red
8 blue
4 redOR
4 blue
have
5 4 8 4
5! 8!
(5 4)! 4! (8 4)! 4!C C
OR ADD
wantwant
An jar contains 13 marbles – 5 red and 8 blue. If four are selected at random, how many ways can you select 4 red marbles or 4 blue marbles?
Jeff Bivin -- LZHS
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?
# of successtotal # of outcomes
Probability – “or”
Pr(3R or 3B) = 5C3 + 8C3
13C3=
5 red
8 blue
have want
3 red
Total: 13 3
13
3
286
66
286
5610
3 blue
want
OR
Jeff Bivin -- LZHS
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?
The AND factor.
AND MULTIPLY
!1)!15(
!5
!2)!29(
!91529
CC
180536
Jeff Bivin -- LZHS
3B2NB or 4B1NB or 5B
At least
3 or 4 or 5 blue
591114921139 CCCCC 126111265584
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?
6132
Jeff Bivin -- LZHS
0R5Nr or 1R4NR
At most
0 or 1 red
4151551505 CCCC 1365530031
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?
9828
Jeff Bivin -- LZHS
PROBABILITY
Definition:
number of successtotal number of outcomes
The ratio
Jeff Bivin -- LZHS
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 successtotal number of outcomes
Pr(H) = 12=
Jeff Bivin -- LZHS
number of success
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:
total number of outcomesPr(>4) =
26=
13=
{ 1 , 2 , 3 , 4 , 5 , 6 }
Jeff Bivin -- LZHS
number of failures
total number of outcomes
total number of outcomesnumber of success
Probability – Success & Failure
A die is tossed, what is the probability that you will obtain a number greater than 4?
Pr(>4) = 26=
13=
What is the probability that you fail to obtain a number greater than 4?
Pr(>4) = 46
23= =
TOTAL = Pr(success) + Pr(failure) = 1
Jeff Bivin -- LZHS
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?
number of successtotal number of outcomes
Probability
Pr(3R) = 5C3
13C3=
5 red
8 blue
have want
3 red
Total: 13 3
143
5
286
10
Jeff Bivin -- LZHS
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?
number of successtotal number of outcomes
Probability
Pr(3B) = 8C3
13C3=
5 red
8 blue
have want
3 blue
Total: 13 3
143
28
286
56
Jeff Bivin -- LZHS
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?
number of successtotal number of outcomes
Probability – “and”
Pr(1R2B) = 5C1 ● 8C2
13C3=
5 red
8 blue
have want
1 red
Total: 13 3
143
70
286
140
286
285
2 blue
multiply
Jeff Bivin -- LZHS
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?
# of successtotal # of outcomes
Pr(1R,1B,1W) = 5C1●8C1●7C1
20C3=
5 red8 blue7 white
have want
1 red
Total: 20 3
57
14
1140
280
1140
785
1 blue
1 white
1 red, 1 blue, & 1 white
Jeff Bivin -- LZHS
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?
# of successtotal # of outcomes
Pr(2R,2W) = 7C2 ● 3C2
15C4=
7 red5 blue3 white
have want
2 red
Total: 15 4
65
3
1365
63
1365
321
2 white
Jeff Bivin -- LZHS
Five cards are dealt from a standard deck of cards. What is the probability that 3 hearts and 2 clubs are obtained?
# of successtotal # of outcomes
Pr(3H,2C) = 13C3 ● 13C2
52C5=
13 diamonds13 hearts13 clubs
13 spades
have want
3 hearts
Total: 52 5
649740
5577
2598960
22308
2598960
78286
2 clubs
Jeff Bivin -- LZHS
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?
# of successtotal # of outcomes
Probability – “or”
Pr(3R or 3B) = 5C3 + 8C3
13C3=
5 red
8 blue
have want
3 red
Total: 13 3
13
3
286
66
286
5610
3 blue
want
OR
Jeff Bivin -- LZHS
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?
5C3 + 8C3 + 7C3 # of success
total # of outcomesPr(3R or 3B or 3w) = 20C3
=
5 red8 blue
7 yellow
have want
3 red
Total: 20 3
286
101
286
355610
3 blue
want
OR
3 red or 3 blue or 3 yellow ?
want
3 yellowOR
Jeff Bivin -- LZHS
26C2 + 4C2 – 2C2 # of successtotal # of outcomes
Probability – “or” with overlap
Pr(2R or 2B) = Pr(2R) + Pr(2K) – Pr(2RK)
52C2=
26 red26 black
have want
2 red
Total: 52 2
2 kings
want
OR
2 red kings
overlap 1326
16325
221
55
1326
330
If two cards are selected from a standard deck of cards, what is the probability that both are red or both are kings?
4 kings48 other
Jeff Bivin -- LZHS
5C2● 8C1 + 5C1 ● 8C2# of success
total # of outcomes
Probability – “and” with “or”
Pr(2R1B or 1R2B) = 13C3
=
5 red
8 blue
have want
2 red
Total: 13 3
13
10
286
220
286
285810
1 blue
want
OR1 red
2 blue
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?
Jeff Bivin -- LZHS
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?
5C2● 8C1 + 5C3# of success
total # of outcomes
Probability – “at least”
Pr(at least 2Red) = 13C3
=
5 red
8 blue
have want
2 red
Total: 13 3
143
45
286
90
286
10810
1 blue
want
OR3 red
2 red or 3 red2 red and 1 blue or 3 red
Pr(2R1B or 3R) =
Jeff Bivin -- LZHS
5C1● 8C2 + 5C2 ● 8C1 + 5C3
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?
Probability – “at least”
Pr(at least 1Red) = 13C3
5 red
8 blue
have want
1 red
Total:13 3
286
10810285
2 blue
want
OR2 red
Pr(1R2B or 2R1B or 3R) =
want
OR3 red
1 blue
143
115
286
230
Jeff Bivin -- LZHS
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
Probability – “at least”
Pr(0R3B) = 13C3
5 red
8 blue
have want
Total:13 3
143
28
286
56
3 blue
In the previous example we found
143
1151Pr red
Pr(success) + Pr(failure) = 1
Jeff Bivin -- LZHS
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?
Probability – “at least”
Pr(>1 red) = 1 – Pr( 0 red )
143
115
286
230
286
5611
313
38 C
C
Pr(success) + Pr(failure) = 1
Pr(success) = 1 - Pr(failure)
Pr(3 blue)
Jeff Bivin -- LZHS
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?
Probability – “at least”
Pr(at least 1Red)
Pr(1R6B or 2R5B or 3R4B or 4R3B or 5R2B or 6R1B or 7R)
Pr(0Red) Pr( 0R7B )
success
failure
FASTEST
Pr(at least 1Red) = 1 - Pr(0R7B) = 717
791C
C
4862
4853
19448
361
Jeff Bivin -- LZHS
720
51228612187121C
CCCCC
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?
Probability – “at least”
Pr(> 3Red) Pr(3-7 red)
Pr(< 3Red) Pr(0-2 red)
success
failure
FASTEST
1 - Pr(0R7NR or 1R6NR or 2R5NR)
77520
303601
646
393
Jeff Bivin -- LZHS
Probability – “with replacement”
2197320
138
138
135
138
138
135
Must use fractions! R B B
Note: In this example
an order is specified
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?
Jeff Bivin -- LZHS
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?
Probability – “with replacement”
2197960
138
138
135
138
138
135
23 3 C
Must use fractions!
Must account of any order!
Problem:
Fractions imply order! R B B