SBM Chapter 7 Probability: Living with the Odds.

1
SBM Chapter 7 Probability Living with the
Odds.

• 7E Counting and Probability
• A Brief Review Factorials

2
First of All.

• You must understand Factorials and Permutations
  Combinations.
• Please read very carefully (page 450) on
  factorial review.
• Then, read very carefully on what permutations
  are(on page 448)
• Then, read very carefully on what combinations
  are(on page 451)

Please Look at Example 1 on page 447-48
3
Factorials...

• Whenever a positive integer n is multiplied by
  all the preceding positive integers, the result
  is called n factorial and is denoted by the
  symbol n!
• This is read as n factorial
• 1! 1
• 2! 2 X 1 2
• 3! 3 X 2 X 1 6
• 4! 4 X 3 X 2 X 1 24
• etc

4
Example One...

• Calculate the following?

5 X 4
5!
5 X 4 X 3 X 2 X 1



3!
3 X 2 X 1
1
20
5
Example Two...

• A brand of ballpoint pen comes in five colors,
  with fine or regular point, and with standard,
  deluxe, or executive styling. How many different
  versions does the pen come in?

5 X 2 X 3 30 versions
6
Example Three...

• A seven-character computer password can be any
  three letters of the alphabet, followed by two
  numerical digits, followed by two more letters.
  How many different passwords are possible?

26 X 26 X 26 X 10 X 10 X 26 X 26
1,188,137,600
7
Permutations...

• Mathematically, we are dealing with permutations
  whenever all selections come from a single group
  of items.
• no item may be selected more than once.
• the order of arrangement matters.
• The total number of permutations possible with a
  group of n items is n!
• where n! n X (n X 1) X X 2 X 1

8
The Permutations Formula...

• If we make r selections from a group of n items,
  the number of permutations is
• where is read as the number of
  permutations of n items taken r at at time.

n !
nPr
(n - r)!
nPr
Please Look at Examples 2-4 on pages 449-51
9
Example Four...

• From a normal deck of 52 playing cards, three
  cards are drawn and placed face up on a table,
  left to right. How many possible results are
  there of this procedure?

n !
52 !
nPr

(n - r)!
(52 - 3)!

132,600 possible results
10
Example Five...

• Ten finalists in a talent show must give final
  performances. Five contestants will perform on
  the first night of the show. How many different
  ways can the schedule for the first night be made?

n !
10 !
nPr

(n - r)!
(10 - 5)!

30,240 different ways
11
Combinations...

• Mathematically, we are dealing with combinations
  whenever all selections come from a single group
  of items.
• no item may be selected more than once
• the order of arrangement does not matter
• For example, ABCD is considered the same as DCBA.

12
The Combinations Formula...

• If we make r selections from a group of n items,
  the number of possible combinations is
• where is read the number of
  combinations of n items taken r at a time.

nPr
n !
nCr

(n - r)! X r !
r !
nCr
Please Look at Examples 5-7 on pages 453-54
13
Example Six...

• How many different ways are there to order a
  medium two-topping pizza, given that there are
  nine toppings to choose from?

nCr
n !
(n - r)! X r !
9 !
9 !



36 Orders
(9 - 2)! X 2 !
7 ! X 2 !
14
Example Seven...

• A scholar is choosing six books to take on
  vacation, from a stack of 34. How many different
  combinations of books are there?

nCr
n !
(n - r)! X r !
34 !
34 !



(34 - 6)! X 6 !
28 ! X 6 !
1,344,904 Combinations
15
Probability Coincidence...

• Coincidences are bound to happen.
• Although a particular outcome may be highly
  unlikely, some similar outcomes may be extremely
  likely or even certain to occur.
• In general, this means that coincidences will
  inevitably be experienced by someone, even if
  these coincidences have a low probability for any
  particular reason.

Please Look at Examples 8-9 on page 455
16
Homework

• 7E Part I
• s 1, 3 - 14
• Part II
• s 15 - 20a, 22a, 22c,
• 23 - 29, 31, 33