WarmUp

1
Warm-Up
1) In which year(s) did the team lose more games
than they won? 2) In which year did the team play
the most games? 3) In which year did the team
play ten games?
10
8
6
Number of Games
4
2
0
1
2
3
4
Year
2
Math I
UNIT QUESTION How do you use probability to make
plans and predict for the future? Standard
MM1D1-3 Todays Question What is a permutation
and how do we use it to solve statistic
problems? Standard MM1D1.b.
3
Probability

• Math I

4
Lets work on some definitions

• Experiment- is a situation involving chance
  that leads to results called outcomes.
• An outcome is the result of a single trial of an
  experiment
• An event is one or more outcomes of an
  experiment.
• Probability is the measure of how likely an event
  is.

5
Probability of an event

• The probability of event A is the number of ways
  event A can occur divided by the total number of
  possible outcomes.
• P(A)The number of ways an event can occur
• Total number of possible outcomes

6
Probability
If P 0, then the event _______ occur.
cannot
It is ________
impossible
If P 1, then the event _____ occur.
must
It is ______
certain
So probability is always a number between ____
and ____.
1
0
7
Complements
All of the probabilities must add up to 100 or
1.0 in decimal form.
Example Classroom P (picking a boy)
0.60 P (picking a girl) ____
0.40
8
A glass jar contains 6 red, 5 green, 8 blue and 3
yellow marbles.Experiment A marble chosen at
random.

• Possible outcomes choosing a red, blue, green or
  yellow marble.
• Probabilities
• P(red) number of ways to choose red 6 3
• total number of marbles
  22 11
• P(green) 5/22, P(blue) ?, P(yellow) ?

9
Ex.
You roll a six-sided die whose sides are numbered
from 1 through 6. What is the probability of
rolling an ODD number?
There are 3 ways to roll an odd number 1, 3, 5.
P
10
Tree Diagrams

• Tree diagrams allow us to see all possible
  outcomes of an event and calculate their
  probabilities.
• This tree diagram shows the probabilities of
  results of flipping three coins.

11
1. Your school cafeteria offers chicken or tuna
sandwiches chips or fruit and milk, apple
juice, or orange juice. If you purchase one
sandwich, one side item and one drink, how many
different lunches can you choose?
There are 12 possible lunches.
Sandwich(2) Side Item(2) Drink(3) Outcomes
chicken tuna
12
Multiplication Counting Principle

• At a sporting goods store, skateboards are
  available in 8 different deck designs. Each deck
  design is available with 4 different wheel
  assemblies. How many skateboard choices does the
  store offer?

32
13
Multiplication Counting Principle

• A father takes his son Tanner to Wendys for
  lunch. He tells Tanner he can get the 5 piece
  nuggets, a spicy chicken sandwich, or a single
  for the main entrée. For sides he can get
  fries, a side salad, potato, or chili. And for
  drinks he can get milk, coke, sprite, or the
  orange drink. How many options for meals does
  Tanner have?

48
14
Many mp3 players can vary the order in which
songs are played. Your mp3 currently only
contains 8 songs (if youre a loser). Find the
number of orders in which the songs can be played.
There are 40,320 possible song orders.
1st Song 2nd 3rd 4th 5th
6th 7th 8th Outcomes
In this situation it makes more sense to use the
Fundamental Counting Principle.

8
40,320
The solution in this example involves the product
of all the integers from n to one (n is
representing the starting value). The product of
all positive integers less than or equal to a
number is a factorial.
15
Factorial
EXAMPLE with Songs eight factorial
8! 8 7 6 5 4 3 2 1 40,320
16
Factorial

• Simplify each expression.
• 4!
• 6!
• c. For the 8th grade field events there are five
  teams Red, Orange, Blue, Green, and Yellow.
  Each team chooses a runner for lanes one through
  5. Find the number of ways to arrange the runners.

4 3 2 1 24
6 5 4 3 2 1 720
5! 5 4 3 2 1 120
17
5. The student council of 15 members must choose
a president, a vice president, a secretary, and a
treasurer.
There are 32,760 permutations for choosing the
class officers.
President Vice Secretary Treasurer
Outcomes
In this situation it makes more sense to use the
Fundamental Counting Principle.

12

15

14
13

32,760
18
Lets say the student council members names
were Hunter, Bethany, Justin, Madison, Kelsey,
Mimi, Taylor, Grace, Maighan, Tori, Alex, Paul,
Whitney, Randi, and Dalton. If Hunter, Maighan,
Whitney, and Alex are elected, would the order in
which they are chosen matter?
President Vice President
Secretary Treasurer
Is Hunter Maighan Whitney Alex
the same as
Whitney Hunter Alex
Maighan?
Although the same individual students are listed
in each example above, the listings are not the
same. Each listing indicates a different student
holding each office. Therefore we must conclude
that the order in which they are chosen matters.
19
Permutation
When deciding who goes 1st, 2nd, etc., order is
important.
A permutation  is an arrangement or listing of
objects in a specific order. 
The order of the arrangement is very important!! 
The notation for a permutation       nPr
n  is the total number of
objects
r is the number of objects selected (wanted)
Note  if  n r   then   nPr    n!
20
Permutation
Notation
21
Permutations

• Simplify each expression.
• a. 12P2
• b. 10P4
• c. At a school science fair, ribbons are given
  for first, second, third, and fourth place, There
  are 20 exhibits in the fair. How many different
  arrangements of four winning exhibits are
  possible?

12 11 132
10 9 8 7 5,040
20P4 20 19 18 17 116,280