Transcript Probability - MAthMakesSense2

Chapter 5
Probability
5.1
Probability of Simple Events
Probability:
Probability:
Probability is the branch of mathematics
which studies “randomness.”
Probability:
Probability is the branch of mathematics
which studies “randomness.”
Probability was first formally developed in
the middle of the 17th century by the French
mathematicians Fermat and Pascal.
Probability:
Probability is the branch of mathematics
which studies “randomness.”
Probability was first formally developed in
the middle of the 17th century by the French
mathematicians Fermat and Pascal.
Chevalier De Mere asked Pascal to
determine the chances of winning various
games of chance (dice, cards, and roulette.)
Probability was developed from these
beginnings over the 18th and 19th centuries.
Probability was developed from these
beginnings over the 18th and 19th centuries.
The mathematical foundations of the subject
were formalized by Kolmogorov and other
Soviet mathematicians in the middle of the
20th century.
Probability was developed from these
beginnings over the 18th and 19th centuries.
The mathematical foundations of the subject
were formalized by Kolmogorov and other
Soviet mathematicians in the middle of the
20th century.
Probability provides the theoretical basis for
statistical practice.
Why use probability?
Why use probability?
Many events can’t be “predicted” before
they happen.
Why use probability?
Many events can’t be “predicted” before
they happen.
That is, we don’t know what the outcome of
the event will be before it happens.
Why use probability?
Many events can’t be “predicted” before
they happen.
That is, we don’t know what the outcome of
the event will be before it happens.
•Will a given person develop lung cancer in the course
of his lifetime?
Why use probability?
Many events can’t be “predicted” before
they happen.
That is, we don’t know what the outcome of
the event will be before it happens.
•Will a given person develop lung cancer in the course
of his lifetime?
•Will this person develop lung cancer if he smokes
three packs a day?
This last example points out how the study
of “randomness” might be a useful thing.
This last example points out how the study
of “randomness” might be a useful thing.
Just because an event is random doesn’t
mean we can’t make some type of judgment
about the “chances” of it occurring.
This last example points out how the study
of “randomness” might be a useful thing.
Just because an event is random doesn’t
mean we can’t make some type of judgment
about the “chances” of it occurring.
•The chances someone who does not smoke will
develop lung cancer in the course of his lifetime is
about 1 in 200.
This last example points out how the study
of “randomness” might be a useful thing.
Just because an event is random doesn’t
mean we can’t make some type of judgment
about the “chances” of it occurring.
•The chances someone who does not smoke will
develop lung cancer in the course of his lifetime is
about 1 in 200.
•The chances someone who smokes will develop lung
cancer is 14 in 200.
Example: There are 30 MLB teams
Example: There are 30 MLB teams
Number the teams from 1 to 30.
Example: There are 30 MLB teams
Number the teams from 1 to 30.
At the start of the season a number between 1 and 30
is randomly selected.
Example: There are 30 MLB teams
Number the teams from 1 to 30.
At the start of the season a number between 1 and 30
is randomly selected.
•What is the probability that the team that has been
selected wins the World Series?
Example: There are 30 MLB teams
Number the teams from 1 to 30.
At the start of the season a number between 1 and 30
is randomly selected.
•What is the probability that the team that has been
selected wins the World Series?
•What is the probability if you learn that the team that
has been selected is the NY Yankees?
Thus even though an event is random, this
does not mean we can’t say anything about
it.
Thus even though an event is random, this
does not mean we can’t say anything about
it.
We can make judgments about how likely or
unlikely is it to happen.
Thus even though an event is random, this
does not mean we can’t say anything about
it.
We can make judgments about how likely or
unlikely is it to happen.
•This is what sports odds-makers do when they give
odds for teams to win or lose.
Thus even though an event is random, this
does not mean we can’t say anything about
it.
We can make judgments about how likely or
unlikely is it to happen.
•This is what sports odds-makers do when they give
odds for teams to win or lose.
The more information we have about a
situation the better we can judge how likely
something is to happen.
Example: What is the safest form of travel?
Example: What is the safest form of travel?
•Auto
Example: What is the safest form of travel?
•Auto
•Train
Example: What is the safest form of travel?
•Auto
•Train
•Bus
Example: What is the safest form of travel?
•Auto
•Train
•Bus
•Airplane
Example: What is the safest form of travel?
Deaths per 100 million miles
•Auto
•Train
•Bus
•Airplane
Example: What is the safest form of travel?
Deaths per 100 million miles
•Auto
•Train
•Bus
•Airplane
,94
Example: What is the safest form of travel?
Deaths per 100 million miles
•Auto
,94
•Train
.04
•Bus
•Airplane
Example: What is the safest form of travel?
Deaths per 100 million miles
•Auto
,94
•Train
.04
•Bus
.02
•Airplane
Example: What is the safest form of travel?
Deaths per 100 million miles
•Auto
,94
•Train
.04
•Bus
.02
•Airplane
.01
Example: What is the safest form of travel?
Deaths per 100 million miles
•Auto
,94
•Train
.04
•Bus
.02
•Airplane
.01
You are 94 times more likely to die from an auto
accident than from a plane crash.
Example: What is the safest form of travel?
Deaths per 100 million miles
•Auto
,94
•Train
.04
•Bus
.02
•Airplane
.01
You are 94 times more likely to die from an auto
accident than from a plane crash.
We can adjust our actions based on how likely an
event is given a certain course of action.
Research studies involving statistics never
determines anything with absolute 100%
“certainty.”
Research studies involving statistics never
determines anything with absolute 100%
“certainty.”
What these studies attempt to do is determine the
probability with which certain events may or may
not occur.
Research studies involving statistics never
determines anything with absolute 100%
“certainty.”
What these studies attempt to do is determine the
probability with which certain events may or may
not occur.
Research often involves determining the effect of
some explanatory variable on the probability that
some outcome variable will happen.
Example:
Example:
Smoking increases the probability of developing
lung cancer by a factor of 14.
Example:
Smoking increases the probability of developing
lung cancer by a factor of 14.
Saying “smoking causes lung cancer” is
misleading: the vast majority of smokers will never
develop lung cancer.
Example:
Smoking increases the probability of developing
lung cancer by a factor of 14.
Saying “smoking causes lung cancer” is
misleading: the vast majority of smokers will never
develop lung cancer.
However, there is a very strong connection
between smoking and lung cancer. Roughly 90%
of all people with lung cancer are smokers.
Example:
Smoking increases the probability of developing
lung cancer by a factor of 14.
Saying “smoking causes lung cancer” is
misleading: the vast majority of smokers will never
develop lung cancer.
However, there is a very strong connection
between smoking and lung cancer. Roughly 90%
of all people with lung cancer are smokers.
Smoking greatly increases the probability of
getting lung cancer.
(Mathematical) Probability is a numerical
measure of the likelihood of a random event.
(Mathematical) Probability is a numerical
measure of the likelihood of a random event.
For one instance of a given random event
we cannot be sure if it is going to happen or
not.
(Mathematical) Probability is a numerical
measure of the likelihood of a random event.
For one instance of a given random event
we cannot be sure if it is going to happen or
not.
However, probability gives the long-term or
large sample proportion with which certain
outcomes will occur.
One Definition of Probability:
One Definition of Probability:
Perform an experiment and observe a random
event.
One Definition of Probability:
Perform an experiment and observe a random
event.
Either a certain outcome occurs or it does not
occur.
One Definition of Probability:
Perform an experiment and observe a random
event.
Either a certain outcome occurs or it does not
occur.
We keep repeating the experiment over and
over again and calculate the percentage of
times out of all the experiments that this event
occurs.
One Definition of Probability:
Perform an experiment and observe a random
event.
Either a certain outcome occurs or it does not
occur.
We keep repeating the experiment over and
over again and calculate the percentage of
times out of all the experiments that this event
occurs.
The long-term proportion (%) with which a
certain outcome is observed is the probability
of that outcome.
Example:
Example:
Experiment: flip a quarter.
Example:
Experiment: flip a quarter.
Event: determine whether the coin lands “heads.”
Example:
Experiment: flip a quarter.
Event: determine whether the coin lands “heads.”
Suppose I perform this “experiment” 10 times and get
the following outcomes:
Example:
Experiment: flip a quarter.
Event: determine whether the coin lands “heads.”
Suppose I perform this “experiment” 10 times and get
the following outcomes: H T T H H T T H T T.
Example:
Experiment: flip a quarter.
Event: determine whether the coin lands “heads.”
Suppose I perform this “experiment” 10 times and get
the following outcomes: H T T H H T T H T T.
The proportion of events which are heads is 4/10 = 0.4.
Example:
Experiment: flip a quarter.
Event: determine whether the coin lands “heads.”
Suppose I perform this “experiment” 10 times and get
the following outcomes: H T T H H T T H T T.
The proportion of events which are heads is 4/10 = 0.4.
For any one flip of the coin, I cannot be sure whether it
will show up heads or tails.
Example:
Experiment: flip a quarter.
Event: determine whether the coin lands “heads.”
Suppose I perform this “experiment” 10 times and get
the following outcomes: H T T H H T T H T T.
The proportion of events which are heads is 4/10 = 0.4.
For any one flip of the coin, I cannot be sure whether it
will show up heads or tails.
However, the long-term proportion should be ½.
The Law of Large Numbers
The Law of Large Numbers
As the number of repetitions of a random
experiment increases. . .
The Law of Large Numbers
As the number of repetitions of a random
experiment increases. . .
. . .the proportion with which a certain outcome
is observed gets closer to the probability of the
outcome.
In probability, an experiment is any
process that can be repeated in which
the results are uncertain.
A simple event is any single outcome
from a probability experiment. Each
simple event is denoted ei.
To make the ideas in the examples more
precise, we need some definitions.
To make the ideas in the examples more
precise, we need some definitions.
The definitions apply to the simplest
situations where probability applies.
To make the ideas in the examples more
precise, we need some definitions.
The definitions apply to the simplest
situations where probability applies.
E.g., flipping coins, card games, roulette,
etc.
To make the ideas in the examples more
precise, we need some definitions.
The definitions apply to the simplest
situations where probability applies.
E.g., flipping coins, card games, roulette,
etc.
We will look at these kinds of situations
first.
DEFINITIONS:
DEFINITIONS:
A random experiment is any process
that can be repeated in which the results
are uncertain.
DEFINITIONS:
A random experiment is any process
that can be repeated in which the results
are uncertain.
A simple event is any single outcome
from a probability experiment.
DEFINITIONS:
A random experiment is any process
that can be repeated in which the results
are uncertain.
A simple event is any single outcome
from a probability experiment.
Each simple event is denoted ei.
Sample space, S, of an
experiment:
Sample space, S, of an
experiment:
The collection of all possible
simple events.
Sample space, S, of an
experiment:
The collection of all possible
simple events.
In other words, the sample space
is a list of all possible outcomes of
a probability experiment.
An event is any collection of
outcomes from a probability
experiment.
An event is any collection of
outcomes from a probability
experiment.
An event consists of one or more
simple events.
An event is any collection of
outcomes from a probability
experiment.
An event consists of one or more
simple events.
Events are denoted using capital
letters such as E.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of flipping a coin.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of flipping a coin.
The sample space S is:
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
The simple events are:
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
The simple events are:
(1) Coin lands Heads
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
The simple events are:
(1) Coin lands Heads
(2) Coin lands Tails
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
The simple events are:
(1) Coin lands Heads
(2) Coin lands Tails
An event E might be:
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
The simple events are:
(1) Coin lands Heads
(2) Coin lands Tails
An event E might be: Coin lands Heads or Tails
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are:
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5, roll a 6,
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5, roll a 6,
An event E:
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5, roll a 6,
An event E: roll an even number
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5, roll a 6,
An event E: roll an even number {2,4,6}.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5, roll a 6,
An event E: roll an even number {2,4,6}.
Another event E:
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5, roll a 6,
An event E: roll an even number {2,4,6}.
Another event E: roll a number greater than 4
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
The simple events are: roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5, roll a 6,
An event E: roll an even number {2,4,6}.
Another event E: roll a number greater than 4 {5,6}.
The probability of an event E, is
the likelihood of that event
occurring.
The probability of an event E, is
the likelihood of that event
occurring.
The probability of E is denoted:
P(E)
Properties of Probabilities
1. The probability of any event E, P(E), must be
between 0 and 1 inclusive.
Properties of Probabilities
1. The probability of any event E, P(E), must be
between 0 and 1 inclusive.
0 < P(E) < 1.
Properties of Probabilities
1. The probability of any event E, P(E), must be
between 0 and 1 inclusive.
0 < P(E) < 1.
2. If an event is impossible, the probability of the
event is 0.
Properties of Probabilities
1. The probability of any event E, P(E), must be
between 0 and 1 inclusive.
0 < P(E) < 1.
2. If an event is impossible, the probability of the
event is 0.
3. If an event is a certainty, the probability of the
event is 1.
Properties of Probabilities
4. If S = {e1, e2, …, en},
Properties of Probabilities
4. If S = {e1, e2, …, en},
P(e1) + P(e2) + … + P(en) = 1.
Properties of Probabilities
4. If S = {e1, e2, …, en},
P(e1) + P(e2) + … + P(en) = 1.
This last property is just another way of saying the
probability that something happens is a
certainty.
An unusual event is an event that has
a probability close to zero.
An unusual event is an event that has
a probability close to zero.
An impossible event has probability
equal to zero.
An unusual event is an event that has
a probability close to zero.
An impossible event has probability
equal to zero.
A likely event has probability close to
one.
An unusual event is an event that has
a probability close to zero.
An impossible event has probability
equal to zero.
A likely event has probability close to
one.
A certain event has probability equal
to one.
The book mentions three methods for
determining the probability of an event:
The book mentions three methods for
determining the probability of an event:
(1)the classical method
The book mentions three methods for
determining the probability of an event:
(1)the classical method
Will cover this in more detail – Pascal and Fermat
The book mentions three methods for
determining the probability of an event:
(1)the classical method
Will cover this in more detail – Pascal and Fermat
(2) the empirical method
The book mentions three methods for
determining the probability of an event:
(1)the classical method
Will cover this in more detail – Pascal and Fermat
(2) the empirical method
Essentially consists of running a random experiment
several times and measuring the proportion of times
each outcome occurs.
The book mentions three methods for
determining the probability of an event:
(1)the classical method
Will cover this in more detail – Pascal and Fermat
(2) the empirical method
Essentially consists of running a random experiment
several times and measuring the proportion of times
each outcome occurs.
(3) the subjective method
The book mentions three methods for
determining the probability of an event:
(1)the classical method
Will cover this in more detail – Pascal and Fermat
(2) the empirical method
Essentially consists of running a random experiment
several times and measuring the proportion of times
each outcome occurs.
(3) the subjective method
Judge based on level of confidence that something
will occur – e.g., what sports odds-makers do.
The book mentions three methods for
determining the probability of an event:
(1)the classical method
Will cover this in more detail – Pascal and Fermat
(2) the empirical method
Essentially consists of running a random experiment
several times and measuring the proportion of times
each outcome occurs.
(3) the subjective method
Judge based on level of confidence that something
will occur – e.g., what sports odds-makers do.
This is what humans do instinctually.
The classical method of computing
probabilities requires equally likely
outcomes.
The classical method of computing
probabilities requires equally likely
outcomes.
An experiment is said to have equally
likely outcomes when each simple event
has the same probability of occurring.
Computing Probability Using the Classical Method
Computing Probability Using the Classical Method
•An experiment has n equally likely simple
events.
Computing Probability Using the Classical Method
•An experiment has n equally likely simple
events.
•The number of ways that an event E can occur
is m.
Computing Probability Using the Classical Method
•An experiment has n equally likely simple
events.
•The number of ways that an event E can occur
is m.
Then the probability of E, P(E), is:
Computing Probability Using the Classical Method
•An experiment has n equally likely simple
events.
•The number of ways that an event E can occur
is m.
Then the probability of E, P(E), is:
Computing Probability Using the Classical Method
If S is the sample space of the experiment.
Computing Probability Using the Classical Method
If S is the sample space of the experiment.
N(S) denotes the number of simple events in S.
Computing Probability Using the Classical Method
If S is the sample space of the experiment.
N(S) denotes the number of simple events in S.
N(E) denotes the number of simple events in E.
Computing Probability Using the Classical Method
If S is the sample space of the experiment.
N(S) denotes the number of simple events in S.
N(E) denotes the number of simple events in E.
EXAMPLE: Classical Probability
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
Each simple event is equally likely (fair coin.)
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
Each simple event is equally likely (fair coin.)
P(H) =
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
Each simple event is equally likely (fair coin.)
P(H) = ½
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
Each simple event is equally likely (fair coin.)
P(H) = ½
P(T) = ½
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
Each simple event is equally likely (fair coin.)
P(H) = ½
P(T) = ½
Let E be the event: H or T {H,T}.
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
Each simple event is equally likely (fair coin.)
P(H) = ½
P(T) = ½
Let E be the event: H or T {H,T}.
P(E) = 1.
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
Each simple event is equally likely (fair coin.)
P(H) = ½
P(T) = ½
Let E be the event: H or T {H,T}.
P(E) = 1.
Let E be the event that we get neither H nor T.
EXAMPLE: Classical Probability
Consider the experiment of flipping a coin.
The sample space S is: {Heads, Tails}
Each simple event is equally likely (fair coin.)
P(H) = ½
P(T) = ½
Let E be the event: H or T {H,T}.
P(E) = 1.
Let E be the event that we get neither H nor T.
P(E) = 0.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(4) = 1/6
P(2) = 1/6
P(3) = 1/6
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(4) = 1/6
P(5) = 1/6
P(3) = 1/6
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
Let E be the event roll is even.
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
Let E be the event roll is even. E = {2,4,6}.
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
Let E be the event roll is even. E = {2,4,6}.
P(E) = N(E)/N(S)
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
Let E be the event roll is even. E = {2,4,6}.
P(E) = N(E)/N(S) = 3/6 = 1/2
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
Let E be the event roll is even. E = {2,4,6}.
P(E) = N(E)/N(S) = 3/6 = 1/2
Let E be the event roll greater than 4.
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
Let E be the event roll is even. E = {2,4,6}.
P(E) = N(E)/N(S) = 3/6 = 1/2
Let E be the event roll greater than 4. E = {5,6}.
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
Let E be the event roll is even. E = {2,4,6}.
P(E) = N(E)/N(S) = 3/6 = 1/2
Let E be the event roll greater than 4. E = {5,6}.
P(E) = N(E)/N(S)
N(S) = 6.
EXAMPLE Identifying Events and the Sample
Space of a Probability Experiment
Consider the experiment of rolling a die.
The sample space S = {1, 2, 3, 4, 5, 6}.
P(1) = 1/6
P(2) = 1/6
P(3) = 1/6
P(4) = 1/6
P(5) = 1/6
P(6) = 1/6
Let E be the event roll is even. E = {2,4,6}.
P(E) = N(E)/N(S) = 3/6 = 1/2
Let E be the event roll greater than 4. E = {5,6}.
P(E) = N(E)/N(S) = 2/6 = 1/3
N(S) = 6.
A probability distribution is just the list of
probabilities for each simple event in the sample
space.
A probability distribution is just the list of
probabilities for each simple event in the sample
space.
For instance, the probability distribution of the fair
coin is:
A probability distribution is just the list of
probabilities for each simple event in the sample
space.
For instance, the probability distribution of the fair
coin is:
P(H) = ½
A probability distribution is just the list of
probabilities for each simple event in the sample
space.
For instance, the probability distribution of the fair
coin is:
P(H) = ½
P(T) = ½
A probability distribution is just the list of
probabilities for each simple event in the sample
space.
For instance, the probability distribution of the fair
coin is:
P(H) = ½
P(T) = ½
The probability distribution of the fair die is:
A probability distribution is just the list of
probabilities for each simple event in the sample
space.
For instance, the probability distribution of the fair
coin is:
P(H) = ½
P(T) = ½
The probability distribution of the fair die is:
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6
The classical probability distribution, where each
simple event is equally likely to occur, is often
called the uniform distribution.
The classical probability distribution, where each
simple event is equally likely to occur, is often
called the uniform distribution.
The assumption that each simple event is equally
likely is often not valid in real-life situations.
The classical probability distribution, where each
simple event is equally likely to occur, is often
called the uniform distribution.
The assumption that each simple event is equally
likely is often not valid in real-life situations.
It may be possible to obtain another distribution
from theoretical considerations (binomial, etc.)
The classical probability distribution, where each
simple event is equally likely to occur, is often
called the uniform distribution.
The assumption that each simple event is equally
likely is often not valid in real-life situations.
It may be possible to obtain another distribution
from theoretical considerations (binomial, etc.)
However, if this is not the case we can obtain an
approximation to the probability distribution
empirically.
EXAMPLE Using Relative Frequencies to
Approximate Probabilities
The following data represent the number of
homes with various types of home heating
fuels based on a survey of 1,000 homes.
What is the approximate probability of each heating source?
What is the approximate probability of each heating source?
504/1000 = .504
What is the approximate probability of each heating source?
504/1000 = .504
64/1000 = .064
What is the approximate probability of each heating source?
504/1000 = .504
64/1000 = .064
307/1000 = .307
What is the approximate probability of each heating source?
504/1000 = .504
64/1000 = .064
307/1000 = .307
94/1000 = .094
What is the approximate probability of each heating source?
504/1000 = .504
64/1000 = .064
307/1000 = .307
94/1000 = .094
2/1000 = .002
What is the approximate probability of each heating source?
504/1000 = .504
64/1000 = .064
307/1000 = .307
94/1000 = .094
2/1000 = .002
17/1000 = .017
What is the approximate probability of each heating source?
504/1000 = .504
64/1000 = .064
307/1000 = .307
94/1000 = .094
2/1000 = .002
17/1000 = .017
1/1000 = .001
What is the approximate probability of each heating source?
504/1000 = .504
64/1000 = .064
307/1000 = .307
94/1000 = .094
2/1000 = .002
17/1000 = .017
1/1000 = .001
4/1000 = .004
What is the approximate probability of each heating source?
504/1000 = .504
64/1000 = .064
307/1000 = .307
94/1000 = .094
2/1000 = .002
17/1000 = .017
1/1000 = .001
4/1000 = .004
7/1000 = .007