#### Transcript Probability Intro

```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?
Number of Games
Warm-Up
10
8
6
4
2
0
Won
1
2
Lost
3
Year
4
Math I
Day 48 (10-19-09)
UNIT QUESTION: How do you use
probability to make plans and predict
for the future?
Standard: MM1D1-3
Today’s Question:
How can I find the different ways to
arrange the letters in the word
“PENCIL”?
Standard: MM1D1.b.
Probability
Math I
October 19, 2009
Statistics
Box Plots
Control Charts
Let’s work on some definitions
Experiment- is a situation
to results called outcomes.

In the previous problem, the
experiment is spinning the
spinner.
An outcome is the result of a
single trial of an experiment

The possible outcomes are
landing on yellow, blue, red
or green

One event of this
experiment is landing on
blue.

The probability of landing
on blue is one fourth.
An event is one or more
outcomes of an experiment.
Probability is the measure of
how likely an event is.
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
P(blue)= number of ways to land on blue
total number of colors
Probability
cannot occur.
If P = 0, then the event _______
impossible
It is ________
must occur.
If P = 1, then the event _____
certain
It is ______
So probability is always a number
1
0
between ____
and ____.
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
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)= ?

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.
3
P =
6
1
=
2
Theoretical or experimental?


We can calculate what our probabilities should
be (theoretical values), but that is not always
what happens in a real experiment. We could
spin the spinner and land on the blue sector
every time (experimental values).
That’s not very likely, but it could happen
Favorable outcomes



Suppose you have the four color spinner-(red,
blue, green and yellow. The probability of
spinning a red is ¼, but how many reds should
you get if you spin it 20 times?
20 * ¼ = 5 times , you should theoretically
land on red 5 times in 20 spins.
Does that always happen with the spinnerswhy don’t the values always match what you
expect?
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.
P(tails)
Probability:
Permutations
Use an appropriate method to find the number of
outcomes in each of the following situations:
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)
chips
chicken
fruit
chips
tuna
fruit
Drink(3)
Outcomes
apple juice
orange juice
milk
apple juice
orange juice
milk
chicken, chips, apple
chicken, chips, orange
chicken, chips, milk
chicken, fruit, apple
chicken, fruit, orange
chicken, fruit, milk
apple juice
orange juice
milk
apple juice
orange juice
milk
tuna, chips, apple
tuna, chips, orange
tuna, chips, milk
tuna, fruit, apple
tuna, fruit, orange
tuna, fruit, milk
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
Multiplication Counting Principle
• A father takes his son, Marcus, to Wendy’s
for lunch. He tells Marcus he can get a 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 a frosty,
coke, sprite, or an orange drink. How
many options for meals does Marcus
have?
48
Many mp3 players can vary the order in which songs are played.
Your mp3 currently only contains 8 songs. 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 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
The solutions in examples 3 and 4 involve the product of all
the integers from n to one. The product of all positive integers
less than or equal to a number is a factorial.
Factorial
The product of counting numbers beginning at n and
counting backward to 1 is written n! and it’s called n
factorial.
factorial.
EXAMPLE with Songs
‘eight factorial’
8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
Factorial
Simplify each expression.
a. 4! 4 • 3 • 2 • 1 = 24
b. 6! 6 • 5 • 4 • 3 • 2 • 1 = 720
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. = 5! = 5 • 4 • 3 • 2 • 1 = 120
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(15) Vice(14)
Secretary (13)
Treasurer(12) Outcomes
In this situation it makes more sense to use the
Fundamental Counting Principle.
15 • 14 • 13 • 12 =
32,760
Let’s say the student council members’ names were: John,
Miranda, Michael, Kim, Pam, Jane, George, Michelle, Sandra,
Lisa, Patrick, Randy, Nicole, Jennifer, and Paul. If Michael, Kim,
Jane, and George are elected, would the order in which they are
chosen matter?
President
Is Michael
Vice President
Kim
Secretary
Jane
Treasurer
George
the same as…
Jane
Michael
George
Kim ?
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.
Permutation
Notation
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!!
P
n!
r = (n  r )!
The notation for a permutation:
n
n is the total number of objects
r is the number of objects selected (wanted)
*Note if n = r then nPr = n!
Permutations
Simplify each expression.
a. 12P2 12 • 11 = 132
b. 10P4 10 • 9 • 8 • 7 = 5,040
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?
= 20P4 = 20 • 19 • 18 • 17 = 116,280
Classwork
Practice Workbook Lesson 6.2 - #12-20
Homework
Page 344 #1-6, 25-28
```