Transcript Tech Math 1 (Probility) 53

Probability
53 Fundamental counting principle
52 Factorials
51 Permutations
50 WP: Permutations
49 Combinations
48 WP: Combinations
Quote
It is easier to gain
forgiveness than to
get permission.
Grace Murray Hopper
Puzzle – What is this?
The maker doesn’t want
it, the buyer doesn’t use
it and the user doesn’t
see it.
What is it?
53a Fundamental counting principle
Fundamental Counting Principal = Fancy
way of describing how one would
determine the number of ways a sequence
of events can take place.
53b Fundamental counting principle
You are at your school cafeteria that allows you to choose a
lunch meal from a set menu. You have two choices for the
Main course (a hamburger or a pizza), Two choices of a drink
(orange juice, apple juice) and Three choices of dessert (pie,
ice cream, jello).
12 meals
How many different meal combos can you select?_________
Method one: Tree diagram
Lunch
Hamburger
Apple
Pie
Icecream
Jello
Orange
Pie
Icecream
Jello
Pizza
Apple
Pie
Icecream
Jello
Orange
Pie
Icecream
Jello
53c Fundamental counting principle
Method two:
Multiply number of choices
2 x 2 x 3 = 12 meals
Ex 2: No repetition
During the Olympic 400m sprint, there are 6 runners. How
many possible ways are there to award first, second, and
third places?
1st
2nd
3rd
3 places
____
6 x ____
5 x ____
4 = 120 different ways
53d Fundamental counting principle
Ex 3: With repetition
License Plates for cars are labeled with 3 letters followed by 3
digits. (In this case, digits refer to digits 0 - 9. If a question
asks for numbers, its 1 - 9 because 0 isn't really a number)
How many possible plates are there? You can use the same
number more than once.
___
26 x ___
26 x ___
26 x ___
10 x ___
10 x ___
10 = 17,576,000 plates
Ex 4: Account numbers for Century Oil Company consist
of five digits. If the first digit cannot be a 0 or 1, how many
account numbers are possible?
___
8 x ___
10 x ___
10 x ___
10 x ___
10 = 80,000 different account #’s
53e Fundamental counting principle
We are going to collect data from cars in the student
parking lot.
License place
1
2
3
4
.
.
.
.
.
50
Vehicle color
Factorials - Quote
Space and time are
intimately intertwined and
indissolubly connected
with each other.
Sir William Rowan Hamilton
Factorials - Puzzle
There is a square fountain that has a tree
growing at each corner. I want to turn this
into a piranha pond, but to do that the size
of the fountain needs to be doubled. How
could I do this without digging deeper or
moving a tree and still have a square
fountain?
52a Factorials
5 • 4 • 3 • 2 • 1 = 5! Factorial
7!= 7 • 6 • 5 • 4 • 3 • 2 •1 = 5040
7! 7  6  5  4  3  2  1

 42
5!
5  4  3  2 1
8! 8  7  6  5  4  3  2  1 8  7  6


5!3! 5  4  3  2  1  3  2  1 3  2  1
87
 56
1
Quote
Algebra is but written
geometry, and
geometry is but written
algebra.
Sophie Germain
Puzzle
What are the last
few hairs on a
dogs tail called?
51a Permutations
Permutations = A listing in which order IS important.
Can be written as:
P(6,4)
or
6 P4
P(6,4) Represents the number of ways 6 items can be
taken 4 at a time…..
Or
6 x 5 x 4 x 3 = 360
Find P(15,3) = _____
2730
15 x 14 x 13
Or
6 (6-1) (6-2) (6-3)
51b Permutations - Activity
Write the letters G R A P H on the top of your paper.
Compose a numbered list of different 5 letter Permutations.
-(not necessarily words)
On the bottom of your paper write how many different
permutations you have come up with.
Don’t forget your Name, Date and Period before turning in.
Hint: You may wish to devise a strategy or pattern for
finding all of the permutations before you start.
Quote
Happy is the man who devotes
himself to a study of the heavens
... their study will furnish him
with the pursuit of enjoyments.
Johannes Kepler
Puzzle
From statistical records,
what is the most dangerous
job in America?
50a WP: Permutations
Use the same formula from section 52 to solve these WPs.
Ex1. Ten people are entered in a race. If there are no ties,
in how many ways can the first three places come out?
8 = 720
___
10 x ___
9 x ___
Ex2. How many different arrangements can be made with
the letters in the word LUNCH?
5! or
4 x ___
3 x ___
2 x ___
1 = 120
___
5 x ___
Ex3. You and 8 friends go to a concert. How many
different ways can you sit in the assigned seats?
9! = 362,880
50b WP: Permutations - Activity
On a separate sheet of paper, use only the letters below to
form as many words as possible.
Don’t forget Name, Date and Period.
Mathematics Permutations
1
2
3
4
.
.
.
.
.
50
Quote
In mathematics there are
no true controversies.
Karl Friedrich Gauss (gowse)
Puzzle
How long will a socalled Eight Day
Clock run without
winding?
49a Combinations
Combinations = A listing in which order is NOT important.
Can be written as:
C(3,2)
or
3C2
C(3,2) means the number of ways 3 items can be
taken 2 at a time. (order does not matter)
Ex. C(3,2) using the letters C A T
CA CT AT
Pr
n Cr 
r!
n
n = total
r = What you want
49b Combinations
Pr
n Cr 
r!
n
C(7,2)
n = total
r = What you want
P2

7 C2 
2!
7
7x6
2x1
=
42
2
= 21
Which is not an expression for the number of ways 3 items
can be selected from 5 items when order is not considered?
Quote
Say what you know,
do what you must,
come what may.
Sonya Kovalevsky (co va LEV ski)
WP: Combinations
If you were to take two apples
from three apples, how many
would you have?
48a WP: Combinations
Permutations = Order IS important
P(8,3) = ___
8 x ___
7 x ___
6
= 336
Combinations = Order does not matter
P(8,3)
336
336


 56
C(8,3) =
3!
3 2
6
48b WP: Combinations
Ex1. A college has seven instructors qualified to teach a
special computer lab course which requires two instructors
to be present. How many different pairs of teachers could
there be?
C(7,2) =
76
 21
2!
Ex2. A panel of judges is to consist of six women and three
men. A list of potential judges includes seven women and six
men. How many different panels could be created from this
list?
Women
Men
65 4
 20
C(7,6)
C(6,3) =
7  6 5 4 3 2 7

6  5  4  3  2 1
3  2 1
7*20 = 140  140 choices