# An Introduction to Probability

advertisement ```Math 12 Advanced: An Introduction to Probability – you should probably do this!!!
PAGE 300
Section 5.1 Probability and Quantifying the Outcomes
P( X ) 
The notation P( X ) means the probability of event X occurring.
Number of Ways of X Occurring
Total Number of Possible Outcomes
Example: Rolling a six sided die, the sample space denoted S  {1,2,3,4,5,6} represents all the possible outcomes.
The probability of rolling and even number: P(even) 
3
or 0.5
6
The probability of rolling a 1, 2, 3, 4, 5, or 6: P(1,2,3,4,5 or 6) 
6
or 1
6
Probability ranges
from zero to one!
0
The probability of rolling a 7: P(7)  or 0
6
Example: Suppose you pick one card from a well shuffled deck of 52 playing cards.
What is the probability of drawing a king?
P(King ) 
What is the probability of not drawing a king?
4
1
or
52
13
P( Not a King ) 
48 12
or
52
13
The probability of “drawing a king” and “not drawing a king” are complements of each other!
P( X )  1  P( X )
The complement of X is often written as X .
Homework Questions:
1. Given a well shuffled deck of 52 playing cards, you pick one card, determine the probability of the following:
a) Drawing a 5
b) Drawing a red 5
c) Drawing a face card
d) Drawing the King of Spades
e) Not drawing the King of Spades
f) Drawing a red card
g) Drawing an Ace or Queen
h) Not drawing an Ace or Queen
i) Drawing a 7, 8 or 9
2. A bag contains three black marbles, six red marbles and 6 white marbles. What is the probability of choosing
each of the following:
a) a black marble
b) a white marble
c) a red or a white marble
3. A hockey team has 2 goalies, six defence and nine forwards. Suppose that one of them is drafted.
Calculate each probability:
a) The draftee is a goalie.
b) The draftee is not a forward.
4. Given 2 fair die to roll, determine the sample space of possible sums by completing the chart below. Then
determine the probability of the following:
Die 1
Die 2
1
1
2
3
4
5
6
a) Rolling a sum of 7
b) Rolling a sum of 11
2
c) Not rolling a sum of 11
3
d) Rolling a sum of 1
4
e) Rolling an even sum
5
f) Rolling 2 sixes
6
From page 302 in you text book:
5. text page 302 #5
6. text page 302 #6
7. a) Explain the difference between theoretical probability and experimental probability.
b) Give an example of a situation in which you would rely on experimental probability rather than on
theoretical probability
8. In theory, what is the P(passing a test)? Explain your answer.
9. If the P(passing a test) =
7
8
and 32 students write the test, how many should pass? Would this be an example
of theoretical or experimental probability? Explain.
PAGE 307
Section 5.2 Counting and Probability
Consider 3 cars in a race – a dodge, a ford and a Honda. How many different ways can they finish 1st and 2nd?
Consider the chart below:
First
Dodge
Dodge
Ford
Ford
Honda
Honda
Second
Ford
Honda
Dodge
Honda
Ford
Dodge
Consider the Tree Diagram below:
Dodge
DF
DH
FD
FH
HF
HD
Ford
Honda
Ford
(DF)
Honda
(DH)
Dodge
(FD)
Honda
(FH)
Ford
(HF)
Dodge
(HD)
How many
different
possibilities
are there?
Consider the following situation: Lori is going to the show. She will go with either Lynne or Chris, they will see a
comedy, a horror or a drama and they will order licorice or popcorn or nachos. How many possibilities must she
consider?&times;&times;
Licorice
Popcorn
Comedy
Nachos
Lynne
Chris
One Possibility: Lynne, Comedy and Licorice
Horror
Licorice
Popcorn
Nachos
Drama
Licorice
Popcorn
Nachos
Comedy
Licorice
Popcorn
Nachos
Would this method have worked with the car race above?
Horror
Licorice
Popcorn
Nachos
Drama
Licorice
Popcorn
Nachos
This is called the Fundamental Counting Principle!
So how many possibilities does Lori have to consider in total?
How could we have achieved the answer of “18” more efficiently?
&times;
(number of
first choices)
&times;
(number of
second choices)
=
(number of
third choices)
(total number
of choices)
Homework Questions: page 308 #1 – 9
Applying the Fundamental Counting Principle to Probability

Read through Focus B on page 309: then do questions 10 – 12 on page 310 (should be doing some
multiplying of probabilities)
PAGE 310: Dependent and Independent Events
Dependent event - Event A is dependent of event B if the result of one does affect the result of the other.
Example: A jar contains three green marbles, four purple marbles and a black marble. Given a green marble (event
A) is drawn and not put back in the bag (not replaced) what is the probability of drawing a purple marble (event
B)?
Answer: PPurple  
4
or 0.57
7
There are 4 purple marbles out of now only 7 total.
Independent event - Event A is independent of event B if the result of one does not affect the results of the other.
Example: A jar contains three green marbles, four purple marbles and a black marble.
Given a green marble (event A) is drawn and the put back in the bag (replaced) what is the probability of drawing
a purple marble (Event B)?
Answer: PPurple  
4 1
 or 0.5
8 2
There are 4 purple marbles out of 8 total.
When there is replacement, the event is considered independent and when there is no replacement the event is
considered dependent.
Homework Questions: page 311 #16 - 20
VENN DIAGRAMS - a way to graphically represent sets and set events
Each diagram begins with a rectangle representing the Sample Space S (often called the universal set) and then
each event is represented by a circle.
Complement of Event A ( A )
Event A
U A
U A
B
U A
B
Event A or B
U
Event B
B
Event A and B
U
Applying Venn Diagrams to Probability:
There are 4 people that own a cat AND
a dog – Bill, Jane, Bob and Lori.
Example: 20 people have been asked if they owned a cat, a dog, or both.
Cat
Bill
Jane
Bob
Lori
Helen
Lynn
Tom
Hugh
Chris
Bill
Jane
Bob
Lori
Dog
Erin
Liam
Ashley
Chelsey
Sara
Amanda
Zach
Luke
Cat
U
5
Dog
4
Note:
5  4  8  3  20
8
3
There are 9 people that
own a cat.
How many people own a cat or a dog or both?
There are 3 people that own
neither a cat nor a dog.
How many people own just a dog?
How many people do not own a cat?
If you can answer these questions, then you can determine the probabilities of these events happening.
Example: P(Cat ) 
9
or 0.45
20
Text Book: Page 312 – 315
P ( Not a Cat ) 
11
or 0.55
20
These events are complements and
together total a probability of one!
Start with Investigation 5 and work through until #36 page 315
Important Formulas: P( A or B)  P( A)  P( B)  P( A and B)
P( A or B)  P( A)  P( B)
P( A and B)  P( A)  P( B)
Do you know what the difference is?
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