1. Counting and Probability
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: April 16, 2008

2. Fundamental Counting Principal
How many different meals can be made if 2 main courses, 3 vegetables, and 2 desserts are available?
Let’s choose a
main course 2
M M2
Now choose a
vegetable 3 x
V V V V V V3
Finally choose
A dessert 2 x
D1 D2 D1 D2 D1 D D1 D2 D1 D2 D1 D2 12

3. Linear Permutations 30 29 28 27 * * * 657,720
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. 30P4 Linear Permutations Alternative 1 2 3 4 30 29 28 27 * * *
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 3 4 30 29 28 27 * * *
Try with your
calculator
president
vice-president
secretary
treasurer
657,720
30P4

5. Permutation Formula
657,720

6. 1.5511 x 1025 Linear Permutations 25 24 23 22 21 * * * * . . . 25!
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
x 1025
25!

7. 25P25 1.5511 x 1025 Linear Permutations Alternative 25 24 23 22 21 * *
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
x 1025
25!
25P25

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

11. More Permutations A B C D E F G H I
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 NOT CIRCULAR
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!

13. 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. . .
PROBLEM: Turning it over results in the same outcome.
So, we must divide by 2.

14. More Permutations How many ways can you arrange the letters MATH ?
How many ways can you arrange the letters ABCDEF ?

15. Permutations with Repetition
How many ways can you arrange the letters AAAB?
Divide by 3!
Let’s 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
What is the problem?
Are there any others?
ABAA
BAAA

16. If all were different, how may ways could we arrange 20 items?
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!

17. If all were different, how may ways could we arrange 17 items?
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 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!

18. ! ? ? ? Permutations ORDER Multiply the possibilities Assume the items
are in a straight line ! or
Use the nPr formula
(if no replacement)
Are the items
in a circle? ?
Divide by the number
of items in the circle
Can the item
be turned over? ?
Divide by 2
Are there duplicate
items in your
arrangement? ?
Divide by the factorial of the
number of each duplicated item

19. 33 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)?26 ∙ 25 10 9 8
letter
number
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?26 ∙ 10
letter
number
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.

23. Combinations
NO order
NO replacement
Use the
nCr formula

24. 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

25. Combinations
A plane contains 12 points, no three of which are co-linear. How many different triangles can be formed?
220

26. 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

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 red marbles? 10

28. 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

29. OR  ADD The OR factor. have want want OR
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?
OR  ADD
have
want
want
5 red
6 white
9 blue
3 red OR
3 blue

30. OR  ADD The OR factor. have want want OR
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?
OR  ADD
have
want
want
5 red
8 blue
4 red OR
4 blue

31. Probability – “or” { Pr(3R or 3B) = =
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 success
5C3 + 8C3
Pr(3R or 3B) = =
total # of outcomes
13C3
have
want
want
5 red
8 blue
3 red { OR
3 blue
Total: 

32. AND  MULTIPLY 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

33. At least 3 or 4 or 5 blue 3B2NB or 4B1NB or 5B
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

34. 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

35. PROBABILITY Definition: The ratio  total number of outcomes
number of success
The ratio 
total number of outcomes

36. 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) = =
total number of outcomes 2

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(>4) = = =
total number of outcomes 6 3

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(>4) = = =
total number of outcomes 6 3
What is the probability that you fail to obtain a number greater than 4?
number of failures 4 2
Pr(>4) = = = 6 3
total number of outcomes 1
Pr(success) + Pr(failure) = 1
TOTAL =

39. 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?
number of success
5C3
Pr(3R) = =
total number of outcomes
13C3
have
want
5 red
8 blue
3 red
Total:  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?
number of success
8C3
Pr(3B) = =
total number of outcomes
13C3
have
want
5 red
8 blue
3 blue
Total:  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?
number of success
5C1 ● 8C2
Pr(1R2B) = =
total number of outcomes
13C3
have
want
5 red
8 blue
1 red
2 blue
Total:  3

42. A jar contains 5 red, 8 blue and 7 white marbles
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
# of success
5C1●8C1●7C1
Pr(1R,1B,1W) = =
total # of outcomes
20C3
have
want
5 red
8 blue
7 white
and
1 red
1 blue
and
1 white
Total:  3

43. A jar contains 7 red, 5 blue and 3 white marbles
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 success
7C2 ● 3C2
Pr(2R,2W) = =
total # of outcomes
15C4
have
want
7 red
5 blue
3 white
2 red
and
2 white
Total:  4

44. Five cards are dealt from a standard deck of cards
Five cards are dealt from a standard deck of cards. What is the probability that 3 hearts and 2 clubs are obtained?
# of success
13C3 ● 13C2
Pr(3H,2C) = =
total # of outcomes
52C5
have
want
13 diamonds
13 hearts
13 clubs
13 spades
and
3 hearts
2 clubs
Total:  5

45. Probability – “or” { Pr(3R or 3B) = =
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 success
5C3 + 8C3
Pr(3R or 3B) = =
total # of outcomes
13C3
have
want
want
5 red
8 blue
3 red { OR
3 blue
Total: 

46. A jar contains 5 red and 8 blue marbles and 7 yellow marbles
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 3w) = =
total # of outcomes
20C3
have
want
want {
want
5 red
8 blue
7 yellow
3 red OR OR
3 blue
3 yellow
Total: 

47. 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 2B) = Pr(2R) + Pr(2K) – Pr(2RK)
# of success
26C2 + 4C2 – 2C2 =
total # of outcomes
52C2
have
want
want {
overlap
26 red
26 black
2 red OR
2 red kings
4 kings
48 other
2 kings
Total: 

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: 

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 or 3 red
2 red and 1 blue or 3 red
5C2● 8C1 + 5C3
# of success
Pr(2R1B or 3R) =
Pr(at least 2Red) = =
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: 

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(1R2B or 2R1B or 3R) =
Pr(at least 1Red) =
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
Total:13 

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
Total:13  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(>1 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
success
Pr(at least 1Red)
Pr(1R6B or 2R5B or 3R4B or 4R3B or 5R2B or 6R1B or 7R)
failure
Pr(0Red)
Pr( 0R7B )
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
success
Pr(> 3Red)  Pr(3-7 red)
failure
Pr(< 3Red)  Pr(0-2 red)
1 - Pr(0R7NR or 1R6NR or 2R5NR)

55. 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!