Probability.notebook
September 29, 2012
Probability
Probability is the study of chance.
Example: You roll a fair die. Each face has an equal chance of coming up.
(This means the results are random.)
What are the possible outcomes (results)?
The list of all the possible outcomes is called the sample space. Its symbol is Ω.
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Example: Roll a fair die.
Ω =
What is the probability of rolling...
a) a 5?
b) a prime number?
c) a number less than 7?
d) a 9?
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What is the probability of rolling...
a) a 5? p = 0.17
b) a prime number? p = 0.5
c) a number less than 7? p = 1 This is known as a certain event.
d) a 9? p = 0 This is known as an impossible event.
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Probability is a number that is always...
between 0 and 1 (when expressed as a fraction or decimal), or
between 0 and 100% (when expressed as a percentage).
To calculate simple probability we can use the formula...
p = number of favourable outcomes
number of possible outcomes
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Example: You randomly pick a card from a standard 52­card deck.
Calculate the probability of picking...
a) a black card.
b) a face card.
c) a red queen.
d) a joker.
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Multiplication Rule (Part I)
Sometimes an experiment can be composed of more than one step.
Example: Roll a fair die, then toss a fair coin.
How many outcomes are possible for this experiment?
a) For the die: 6 outcomes
b) For the coin: 2 outcomes
Total number of outcomes = 6 2 = 12 outcomes
* Multiply the number of outcomes for each step of the experiment.
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Example:
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An experiment consists of randomly choosing a card from a standard deck, then tossing a tetrahedral die (4 faces). How many outcomes are possible?
Answer: 52 x 4 = 208 outcomes
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Example:
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Jake has 9 items of clothing that he has to hang on the clothes line. How many different ways can he do this?
Answer: 9 8 7 6 5 4 3 2 1
= 362 880
On calculator: 9! (read 9 factorial) This is called a dependent experiment because the outcomes of some steps depend on the outcomes of previous steps.
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Multiplication Rule (Part II)
Example: Roll a fair die, then randomly choose a card from a standard deck.
a) What are the number of possible outcomes?
b) Is this an independent or dependent experiment?
c) What is the probability of rolling a 4, then picking an ace ?
p(4) =
and
p(ace) = 9
Probability.notebook
p(4) =
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and p(ace) = To calculate the probability of getting both outcomes, we multiply the probabilities of each outcome .
p(4 and ace) = = 10
Probability.notebook
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Showing the Outcomes for a Compound Experiment
Probability tree
Example: A bag contains 9 marbles: 3 red, 2 white and 4 blue. You randomly select 2 marbles, in succession, replacing the first marble. What are all the possible outcomes?
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Example: A bag contains 9 marbles: 3 red, 2 white and 4 blue. You randomly select 2 marbles, in succession, replacing the first marble. What are all the possible outcomes?
Second marble
First marble
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Example: A bag contains 9 marbles: 3 red, 1 white and 5 blue. You randomly select 2 marbles, in succession, without replacing the first marble. What are all the possible outcomes?
Second marble
First marble
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Events
An event can be a single outcome (a simple event) or several outcomes of the sample space.
Example: Draw a card from a standard deck. Drawing a king is an event that corresponds to the outcomes {king of diamonds, king of spades, king of hearts, king of clubs}
Drawing the ace of spades is a simple event because it corresponds to only one of the 52 outcomes.
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An event is said to have occurred if you get any of the outcomes that satisfy that event.
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Example: A box contains 6 marbles: 2 green, 1 yellow and 3 black. Two marbles are drawn in succession and without replacement. What is the probability that both marbles are the same colour?
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The outcomes that satisfy this event 'two of the same colour' are {(G, G), (B, B)}.
To calculate the probability of an event, add the probabilities of all the outcomes that satisfy that event.
p(same colour)
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Two events are called mutually exclusive if they have no outcomes in common.
Example: Event A ­ Drawing a face card
Event B ­ Drawing an ace
If they do have common outcomes, the events are compatible.
Example: Event A ­ Drawing a red card
Event B ­ Drawing an ace
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Two events are called complementary if they have no outcomes in common (mutually exclusive) and together, they make up the sample space.
Example: Event A ­ Drawing a red card
Event B ­ Drawing a black card
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Venn Diagrams
Experiment: Roll a die
Event A: Roll an even number
Event B: Roll a prime number
Event C: Roll an odd number
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Using Venn Diagrams, show
a) Events A and B
A
4
3
B
2
6
5
1
compatible
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b) Events B and C
6
B
2
3
C
1
5
4
compatible
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c) Events A & C
A
2
4
6
C
3
5
1
complementary
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Example: There are 15 marbles in a jar:
7 red, 3 blue and 5 purple.
You choose two randomly and
without replacement.
Event A: 2 same colour
Event B: 2 primary colours
Represent this situation using a Venn
Diagram.
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(P,R)
B
(B,P)
A
(R,R)
(P,P)
(R,B)
(B,B)
(B,R)
(P,B)
(R, P)
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Union and Intersection of Events
1.
Union: All the outcomes of each event are
combined to make one group.
2. Intersection: All the outcomes that the
events have in common.
Example:
Experiment: Roll a die
Event A: Roll an even number
Event B: Roll a factor of 4
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A
B
6
5
2
4
1
Union:
3
Intersection:
Probability of Union & Intersection
Probabilities can be calculated from the
information in the Venn Diagram.
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There is also a formula...
When we put the probability
of the two sets together, the
common outcomes would be
counted twice; therefore we
must subtract the probability
of the intersection to avoid
this.
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The "prime" symbol refers to the
complementary event - in this case, the
outcomes that are "not" part of the
union of events and .
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Warm-up:
A game of chance involves successively drawing two
marbles from a bag. The first marble drawn is not
put back in the bag. The bag contains 12 marbles;
they are either red, green or blue.
The probability of drawing a blue marble followed by
a red marble is
.
Two marbles are randomly drawn from the bag.
What is the probability that the two marbles are
green?
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An experiment consists of rolling a fair die. Consider
the following events:
A: rolling an even number
B: rolling a number less than 4
a)
Construct a Venn diagram that represents this
situation.
b)
Calculate ...
1)
2)
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In the following diagram, the results are equally likely.
B
A
Calculate:
a)
b)
c)
d)
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If , and ,
determine ...
a)
b)
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Conditional Probability
The probability that an event will occur knowing
that another event has already occurred is called
conditional probability.
Example: Roll a fair six-sided die
Event A: Roll and odd number
Event B: Roll a factor of six
What is the probability of rolling an
odd number given that the side facing
up is a factor of six?
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The question is: What is the probability of A
given B.
In symbols we write:
or
.
We can make a diagram:
A
B
=
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We can use a formula:
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Example: In a class of 25 students, we observe
that 18 pass their English test, 16 pass
their math test and 12 pass both tests.
One of these students is selected at
random.
What is the probabilty that...
a) the student passed his English test,
given that he passed his math test?
b) the student passed his math test,
given that he passed his English
test?
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Often a contingency table can be used instead of
a Venn diagram.
Example: 1000 people were surveyed about their
favourite season. The results are shown
in the table.
Women
Men
Fall
65
55
One of these people
Winter
60
90
is chosen at random.
Spring
130
120
Given that the person
Summer
345
135
is a man, what is the
probability that his favourite season is winter?
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You randomly draw a card from a standard 52­card deck.
Event A: Drawing a red card
Event B: Drawing a black card
Event C: Drawing a face card
Calculate:
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Types of Probability
1) Theoretical -- The experiment is never carried out;
the probability is calculated using
reasoning, based on the numbers
involved in the situation.
2) Empirical or Experimental -- The experiment is
actually carried out; the probability
is calculated using the numbers
obtained.
3) Subjective -- The probability is based on the
judgment of people who are
considered experts.
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Odds
Odds for or against an outcome are expressed as ratios.
1)
Odds For
number of favourable outcomes : number of unfavourable outcomes
2)
Odds Against
number of unfavourable outcomes : number of favourable outcomes
Example:
The odds that a hockey team will win their next game are
3:2. (This is an odds for)
This means that the team has 3 chances of winning and 2
chances of losing.
That means that the odds against the team winning are
2:3.
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Odds and Betting
Example:
The odds that a certain horse will win the
Derby are 3:5.
a) Tim places a $200 bet on the horse to
win. How much will he win if the horse
wins the race?
b) Dan bets $100 that the horse will lose.
If the horse does lose the race, then how
much will Dan win?
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Odds and Probability
Example: The odds that James passes his next test are
4:5. What is the probability that he passes the
test?
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Warm-ups:
1. The probability of winning the "Oscar" for a
certain actress is 7/10.
a)
b)
What are the odds for her winning?
You bet $30 that she will win on the
night. How much will you win if the
actress gets the prize?
2. A survey of 200 families living on an island found
that 92 of them owned a helicopter, 75 owned a
hydroplane and 23 owned both. What is the
probability that a randomly selected family...
a)
b)
owns a hydroplane, given that it owns a
helicopter?
owns neither a helicopter nor a hydroplane?
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Mathematical Expectation
* Also known as Expected Value.
* Is a 'weighted average' involving probability.
Example:
A company makes hockey sticks.
These sticks 1) can be sold wholesale, for
a profit of $3 each.
2) can be sold retail, for a
profit of $5 each.
3) can be defective, resulting
in a loss of $15 each.
The company estimates that 12% of their
sticks are rejected, 40% are sold wholesale,
and 48 % are sold retail.
What is the company's expected profit?
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Outcome Probability Net Value
wholesale
retail
defective
Expected Value = Poutcome 1 Voutcome 1 + Poutcome 2 Voutcome 2 + ...
where P represents probability, and V represents net
value of the outcome.
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EV = 49
Probability.notebook
Example:
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A roulette wheel has 37 slots numbered 0
through 36. If you pick a winning number,
you get your money back, plus 35 times the
amount you bet.
Joan places a $20 bet on a number.
a)
b)
How much can Joan expect to win?
Is it worth her while to play this game?
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When the expected value...
1) equals 0, the game is considered fair.
2) is greater than 0 (positive), the game favours the
player.
3) is less than 0 (negative), the game does not favour
the player.
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Warm - ups
1. A game costs $10 to play. You roll a fair die and the
roll determines the amount of money you win.
• If you roll an odd number, you lose your bet.
• If you roll a 2 or a 6, you get your bet back.
• If you roll a 4, you win $20, plus you get your bet
back.
Is this game fair?
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2. You pay $5 and randomly draw a card from a standard
52-card deck.
• If you draw an ace, you win four times your bet.
• if you draw a face card, you win twice your bet.
• Any other outcome, you lose.
Is this game fair?
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Mathematical Expectation: Part II
• The expected value is already known, but the bet or a
prize amount is unknown, and we must work
backwards to find it.
Example: A game of chance involves opening one of nine
doors. Behind these doors are 4 circles, 3
rhombuses and 2 triangles.
Players must bet $5.
* If a circle is revealed, players
lose the money they bet.
* If a rhombus is revealed, players
wins $2 and keeps the money they bet.
* If a triangle is revealed, players win a certain
amount of money and keep the money they
bet.
The game is fair.
How much money does a player win for
choosing a triangle?
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Outcome Probability Net Value
circle
rhombus
triangle
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Example: An amusement park offers a game of chance
that involves spinning a pointer on a wheel that
is divided into 12 congruent
sectors.
If the wheel stops on a
white sector, players win
the amount indicated,
plus they keep the money
they bet. If the wheel stops in
a shaded section, players lose the money they
bet.
This game is fair.
How much money must players bet to play?
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Outcome Probability Net Value
$4
$6
$12
Lose
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Example:
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Another amusement park has the same
game, but this time players do not get
their bet back.
This game is also fair.
How much money must the players bet?
Outcome Probability Net Value
$4
$6
$12
Lose
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