Transcript Powerpoint Lessons

Math 30-2
Probability & Odds
Acceptable Standards
(50-79%)
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The student can
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express odds for or odds against as a probability
determine the probability of an event
determine odds for or odds against an event
determine the odds for an event given the odds against the event and vice versa
distinguish between mutually exclusive events and non-mutually exclusive
events
determine P(A or B) for events that are mutually exclusive
determine P(A) when given P(A or B) and P(B) for mutually exclusive events
interpret a model that represents any combination of mutually exclusive and
non-mutually exclusive events
identify events that are complementary
describe the elements that belong to the complement of a simple event
determine the probability of the complement of an event, given the probability of
the event
create a sample space using a graphic organizer
distinguish between independent and dependent events
determine P(A and B) for independent events
determine P(A and B) for dependent events, given the order of the events
Standards of Excellence
(80% +)
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The student can also
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express probability as odds for or odds against
provide an explanation for the validity of a probability statement
provide an explanation for the validity of an odds statement
determine P(A or B) for events that are non-mutually exclusive
determine P(A) when given P(A or B), P(A and B), and P(B) for
non-mutually exclusive events
represent events that are non-mutually exclusive using a graphic
organizer
describe the elements that belong to the complement of a
compound event
determine P(A) when given P(A and B) and P(B) for independent
events
determine P(A and B) for dependent events when the order of the
events is not given
Vocabulary
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Probability (P) – is the likelihood that an event
will occur.
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Outcomes – when you do a probability
experiment the different possible results are
called outcomes
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Event – is a collection of outcomes
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Sample space: the set of all possible outcomes.
We denote S
Listing the Sample Space
Use a tree diagram to list the sample space for
tossing a coin and rolling a die.
Coin
Die
H
1
2
3
4
5
6
H, 1
H, 2
H, 3
H, 4
H, 5
H, 6
1
2
3
4
5
6
T,
T,
T,
T,
T,
T,
T
Outcomes
1
2
3
4
5
6
The
Sample
Space
Types of Probability
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There are 2 types of probability
Theoretical Probability
Experimental Probability
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Let’s look at each one individually…
Theoretical Probability
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Theoretical Probability is based upon the
number of favorable outcomes divided by
the total number of outcomes
Example:
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In the roll of a die, the probability of getting an
even number is 3/6 or ½.
Theoretical Probability Formula:
Example # 2
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A box contains 5 green pens, 3 blue pens, 8
black pens and 4 red pens. A pen is picked at
random
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What is the probability that the pen is green?
There are 5 + 3 + 8 + 4 or 20 pens in the box
P (green) =
# green pens
Total # of pens
= 5 =1
20 4
Experimental Probability
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As the name suggests, Experimental
Probability is based upon repetitions of
an actual experiment.
Example: If you toss a coin 10 times and record
that the number of times the result was 8 heads,
then the experimental probability was 8/10 or 4/5
Experimental Probability Formula:
P = Number of favorable outcomes
Total number trials
Example
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In an experiment a coin is tossed 15
times. The recorded outcomes were: 6
heads and 9 tails. What was the
experimental probability of the coin being
heads?
P (heads) =
# Heads
Total # Tosses
=
6
15
Complementary
The sum of all the probabilities of an event is equal to 1.
If P = 1, then the event is a certainty.
If P = 0, then the event is impossible.
In probability, if Event A occurs, there is also the probability
that Event A will not occur. Event A not occurring is the
compliment of Event A occurring.
The probability of Event A not occurring is written as P(A).
(This is read as “Probability of not A”).
For Event A: P(A) + P(A) = 1
P(A) = 1 - P(A)
Example
One card is drawn from a deck of 52 cards. What is the
probability of each of these events?
a) drawing a red four b) not drawing a red four
1
a ) P( red 4 ) 
26
1
b ) P( not red 4 )  1 
26
25
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26
Odds
Odds
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Another way to describe the chance of an event occurring is
with odds. The odds in favor of an event is the ratio that
compares the number of ways the event can occur to the
number of ways the event cannot occur.
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We can determine odds using the following ratios:
Odds in Favor =
Odds against =
number of successes
number of failures
number of failures
number of successes
Also can write it as:
odds in favor of A =
number of outcomes for A : number of outcomes against A
Example
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Suppose we play a game with 2 number cubes.
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If the sum of the numbers rolled is 6 or less – you win!
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If the sum of the numbers rolled is not 6 or less – you lose
In this situation we can express odds as follows:
Odds in favor =
Odds against =
numbers rolled is 6 or less
numbers rolled is not 6 or less
numbers rolled is not 6 or less
numbers rolled is 6 or less
Example #2
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A bag contains 5 yellow marbles, 3 white marbles,
and 1 black marble. What are the odds drawing a
white marble from the bag?
Odds in favor =
number of white marbles
number of non-white marbles
3
6
Odds against =
number of non-white marbles
number of white marbles
6
3
Therefore, the odds for are 1:2
and the odds against are 2:1
Practice
Suppose you are watching a game show on TV. Five green
doors are shown. Contestants on the show get to choose a
door and potentially win a prize. Prizes can be found behind
two of the five doors.
1)
2)
3)
4)
Determine the probability of winning a prize.
Determine the probability of not winning a prize.
Add the probability of winning and not winning a prize. What
do you notice?
Use the following formula to write the odds as a fraction.
Write the odds of winning as a ratio. odds in favour of A =
number of outcomes for A : number of outcomes against
6) Write the odds against winning as a ratio.
5)
Discussion
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How does the probability of winning a prize compare to the
odds of winning a prize?
What similarities and differences do you notice between
expressing the probability and the odds for an event?
What do you notice about the odds for winning versus the
odds against winning? What relationship do you see?
Practice 2
There is a 10% probability of winning a free play in the charity draw.
1) Write 10% as a fraction.
2) If 100 tickets are purchased, theoretically how many tickets would
win a free play?
3) If 100 tickets are purchased, theoretically how many tickets would
not win a free play?
4) Based on a 100 tickets being sold, what are the odds in favour of
winning a free play? Write your answer as a ratio; then write it as
a reduced ratio.
5) Describe in words the information you can gain from knowing the
probability of an event and how this information helps you write
the odds for the event.
Probability vs Odds
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Probability is based on winning a prize out of all of the
possibilities, whereas odds are based on winning a prize
compared to not winning a prize.
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The difference between odds and probability is this:
 Probability is based on favourable outcomes in relation
to the total number of possible outcomes.
 Odds are based on the favourable outcomes “for” in
relation to unfavourable outcomes “against.”
Classifying Exclusivity
Mutually Exclusive:
Two events are mutually exclusive if they cannot occur
simultaneously.
For instance, the events of drawing a diamond and drawing a club from
a deck of cards are mutually exclusive because they cannot both occur
at the same time.
U
For mutually exclusive events:
A
B
P(A U B) = P(A) + P(B)
diamond
club
Classifying Exclusivity
Non- Mutually Exclusive
Events that are not mutually exclusive have some common
outcomes.
For instance, the events of drawing a diamond and
drawing a king from a deck of cards are not mutually exclusive
because the king of diamonds could be drawn, thereby having
both events occur at the same time.
U
For events that are not mutually exclusive:
P(A U B) = P(A) + P(B) - P(A and B)
Venn Diagram:
Events that are Not Mutually Exclusive
King
Diamonds
Both king and a diamond
These events are not mutually exclusive as it is possible for
a card to be both king and a diamond
Classifying Exclusivity
Classify each event as mutually exclusive or not mutually exclusive.
a) choosing an even number and choosing a prime number
b) picking a red marble and picking a green marble
c) living in Edmonton and living in Alberta
d) scoring a goal in hockey and winning the game
e) having blue eyes and black hair
Example
1. A box contains six green marbles, four white marbles,
nine red marbles, and five black marbles. If you pick one
marble at a time, find the probability of picking
a) a green or a black marble.
b) a white or a red marble.
Example 2
Determine the probability of choosing a diamond or a face card
from a deck of cards.
Example 3
A national survey revealed that 12.0% of people exercise
regularly, 4.6% diet regularly, and 3.5% both exercise and
diet regularly. What is the probability that a randomly-selected
person neither exercises nor diets regularly?
Work on Practice Questions 1-8
Independent Versus Dependent Events
Two events are independent if the probability that
each event will occur is not affected by the
occurrence of the other event.
If the probabilities of two events are P(A) and P(B)
respectively, then the probability that both events
will occur, P(A and B), is:
P(A and B) = P(A) x P(B) (INDEPENDENT EVENTS)
Conditional Probability
If A and B are events from an experiment, the conditional
Probability is the probability that Event B will occur given that Event A
has already occurred. (dependent event)
P(A Ç B) = P(A)· P(B A)
P(B|A) is the notation for conditional probability.
It should be read as
“the probability of event B happening, given that event A has already occurred.”
A tree diagram is useful for modeling this types of problems
Practice
Two events are dependent if the outcome of the second event is
affected by the occurrence of the first event.
Classify the following events as independent or dependent:
a) tossing a head and rolling a six
b) drawing a face card, and not returning it to the deck, and
then drawing another face card
c) drawing a face card and returning it to the deck, and then
drawing another face card
Example
A cookie jar contains 10 chocolate and 8 vanilla cookies.
If the first cookie drawn is replaced, find the probability of:
a) drawing a vanilla and then a chocolate cookie
b) drawing two chocolate cookies
Example
Find the probability of drawing a vanilla and then drawing a
chocolate cookie, if the first cookie drawn is eaten.
Practice
Determine the conditional probability for each of the following:
a) Given P(A and B) = 0.725 and P(A) = 0.78, find P(B|A).
b) Given P(blonde and tall) = 0.5 and P(B|A) = 0.68, find the P(blonde).
Finding Conditional Probability
The local hockey time is having a raffle to raise money. The team is selling
2500 tickets, and there will be two draws. The first draw is for the grand
Prize—a trip for two to an all-inclusive resort. The second draw is for the
consolation prize-an HDTV. After each draw, the winning ticket is not return
to the raffle. You buy 10 tickets for the raffle.
Calculate the probability of winning the HDTV.
What is the probability of winning at least one prize?
Finding Conditional Probability
A diagnostic test for liver disease is accurate 93% of the time,
and 0.9% of the population actually has liver disease.
a) Determine the probability the patient tests positive
b) Determine the probability the patient tests negative
c) Determine the probability the patient has liver
disease and tests positive
d) Determine the probability the patient does not have
liver disease and tests negative