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Probability

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How many different combinations of cards could be drawn in Blackjack ... One has a new car behind it and the other 2 have a goat. ... – PowerPoint PPT presentation

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Title: Probability


1
Probability
  • The tool that allows the statisticians to use
    sample information to make inferences about or to
    describe the population from which the sample was
    drawn

2
Some concepts
  • Experiment is the process by which an
    observation is obtained.
  • Examples
  • - toss a die is a experiment
  • - Draw a card from a deck of card
  • Event is the outcome of an experiment
  • Examples
  • - toss a die observe a 1
  • observe an odd number
    (1,3,5)

3
Simple Event an event that cannot be decomposed
is called a simple event Examples -toss a die,
event that observe a 1 is a simple event.
event that observe an odd number is not - How
many simple events are there in the card
experiment? Each experiment will result in
one and only one of the simple events. An
event is a collection of one or more simple
events - the event of observing an odd number is
made up of 3 simple events
4
  • Sample Space is the set of all possible
    direct outcomes of an experiment
  • Example
  • - tossing a dice, sample space (1,2,3,4,5,6)
  • - tossing two dice?

5
  • Formal Definition of Probability If you repeat
    an experiment over and over again say (N) times
    and event A is observed n times, then we view
    the probability of event as
  • P(A) n / N
  • practically, we cannot get the probability by
    using this method. One of approaches we are
    going to learn to calculate the probability is
    called Simple Events Approach
  • of simple events included in A
  • P(A)
  • of simple events included in
    sample space.
  • of possible direct outcomes in A
  • P(A)
  • of all possible direct outcomes of
    the experiment

6
  • Property of simple event probability
  • Each probability must lie between 0 and 1,
    inclusive. (any number negative or greater than 1
    is not probability)
  • The sum of the probabilities for all simple
    events in S equals to 1 (the probability of
    sample space is 1)

7
Example Experiment - toss a coin twice (H head
T tail) Step one list all simple event, table
or tree Step two identify the sample space Step
three calculate probabilities What is the
probability that we observe (H,H)
observe the same type observe the different
type observe at least one head
8
Problem toss the coin three times. draw tree
diagram how many simple events can we
have? what is sample space? what is prob of
following events observe (H H T) observe
same type three times. observe at most one
head
9
Learning to Count (all over again)
  • Playing the lottery, how many different possible
    combinations are there for the winning number
  • How many different ways can a 8 person batting
    order be arranged with 9 players
  • How many different combinations of cards could be
    drawn in Blackjack

10
How to count when you are picking r things out of
n
  • Two questions you have to ask
  • Can there be repeats or replacements
  • Does the order matter
  • Results
  • There can be repeats and order matters nr
  • There can not be repeats and order matters
    permutations
  • There can not be repeats and order does not
    matter combination
  • There can be repeats and order does matter not
    talked about yet

11
Combination and Permutation Pick r things out of
n at a time, can not have repeats Permutation
formula If position (order) does matter, we have
following number of choices Combination
formula If position (order) does not matter, we
have following number of choices
12
Intuition of combination and permutation
formula A permutation can be thought of a
selection process in which objects are selected
one by one in a certain order. If we want to
select r objects out of n, the number of first
possible choice is n, and then next is n-1
(because they cannot be the same objects more
than once), and then n-2, finally (n-r1). so the
choices are n(n-1)(n-2)…(n-r1) Given one
combination(select r out of n, no matter what
order they are), we should have r! number of
possible permutations. So the number of
combination is number of permutation divided by
r!
13
  • Problem codes edition
  • Choose 4 numbers from 09 to comprise a 4-digit
    code.
  • If number cannot be the same, how many codes can
    we have?
  • If number can be the same, how many codes can we
    have?

14
  • Problem Cold Stone Ice Cream
  • Suppose Cold Stone provide 7 different types of
    toppings to put in your ice cream
  • If you have enough money to choose three
    different toppings and the order those toppings
    go in does not matter, how many choices do you
    have?
  • If you have enough money to choose 4 different
    toppings and the order those toppings go in does
    not matter, how many choices do you have?

15
  • Problem Coachs choice
  • Ol Roy must choose 5 out of 10 players to put on
    his starting lineup.
  • If players position does not matter, how many
    choices the coach has?
  • If position does matter, how many choices he has?

16
Event composition and Event relations
Intersection of event A and B, denoted by AB or
( ), is the event that both A and B
occur Example toss a die, what is AB for
following 1) A1, Bobserve odd number
2) Aobserve odd number, Beven number
3) Anumber greater than 4, BSample space
17
Event Composition and Relations
  • Union of event A and B, denoted by AB (
    ), is the event that A or B or both occur.
  • Example toss a die, what is AB
  • 1) A1, Bobserve odd number
  • 2) Aobserve odd number, Beven number
  • 3) Anumber greater than 4, BSample space

18
Event Composition and Relations
  • Complement of an event A, denoted by
    consists of the all the simple event in the
    sample space that are not in A.
  • Example given following A, what is A complement
  • 1) A1
  • 2) Aobserve odd number
  • 3) Asample space

19
Event Composition and Relations
  • Mutually exclusive/ Disjoint events two
    events A and B are said to be mutually exclusive
    if when A occurs, B cannot occur(and visa
    versa).
  • This implies that A and B have no common parts
    and empty set( ) )

20
Property of disjoint events
  • 1) P(AB) 0
  • 2) P( ) P(A) P(B)

S
A
B
21
  • Example toss a die
  • Aobserve odd number.
  • Bobserve even number
  • Are they disjoint events?
  • Use simple events approach to calculate P(A),
    P(B), P( ) and P( )

22
  • Additive Rule of Probability
  • - Given two events A and B,
  • P( AB ) P( A ) P( B ) - P( AB )
  • If A and B are mutually exclusive, then P(AB)0
    and
  • P(AB) P ( A ) P ( B )

23
Example toss the coin twice Aobserve at least
one head. Bobserve at least one tail Define
events A, B, AB, and as collection of
simple events. Use simple events approach to
calculate the probability of these events. Use
additive rule of probability to calculate the P(
)
24
Another important of relationship Given any
event A, P(A) P ( ) 1
25
Example toss a die twice What is probability of
sum of two observation greater than 3.
26
Conditional Probability Conditional
probability of event B given A has occurred is
Example toss a die what is probability of
observing a 1 given we already observed an odd
number on the first role
27
Independence two events A and B are said to
be independent if and only if either P(AB)P(A)
or P(BA)P(B) - The fact that B has occurred
has no effect on the probability that A will
occur
28
Multiplicative Rule of probability P(AB)P(A)P(B
A)P(B)P(AB) If A and B are independent events,
P(AB)P(A)P(B) Similarly, if A, B and C are
mutually independent events, then
P(ABC)P(A)P(B)P(C)
29
Problem babys gender a)Two couples plan to have
child this year and what is the probability
of both couples have boys? (Suppose having baby
for each couple are independent events and there
is an equal chance of having a boy or
girl.) b)What is the probability that either
couple or both has a boy? c)If you know one
couple has had a boy what is the probability the
other has had a boy.
30
How to Solve a Problem
  • Read the question and look for the phrase what
    it is the probability of/that _____
  • Decide how to define the event(s) according to
    the blank
  • Decide if you are looking at an intersection
    (and), union (or), conditional (given), or just a
    probability
  • Take the given information and put in in a tree
    or a table, then fill in the blanks
  • Answer the question posed.

31
Tree
  • Multiply up the branches
  • Add down the nodes

32
Probability table for the event A and B there is
a two-way table whose four entries are the four
intersection probabilities P(AB), P(A ), P(
B ), P( ) and whose marginal row and
column sums correspond to the unconditional
probabilities P(A), P( ), P(B) and P( ) as
in the table below B
A P(AB) P(A )
P(A) add across P( B )
P( ) P( ) add across
P(B) P( ) sum down and
across add down add down to
equal 1
33
Problem Is ABS useful? In a survey involving 100
cars, each vehicle was classifies according to
whether or not it has antilock brakes system(ABS)
and whether or not it has been involved in an
accident in the past year. Suppose that any one
of these cars is randomly selected for
inspection. ABS
No ABS Accident 3
12 No Accident
40 45 Transform this
table into a probability table first.
34
  • Questions
  • 1) What is probability that the car has been
    involved in an accident in the past year?
  • 2) What is probability that the car has ABS
  • 3) What is probability that the car has not
    been in an accident and has ABS
  • Given that the car has been involved in an
    accident, what is the probability that it has
    ABS?
  • Given that the car has ABS, what is the prob that
    it has been involved in an accident?
  • Given that the car does not have ABS, what is the
    prob that it has been involved in an accident?

35
Conditional probability and information
update Consider this example first Suppose 100
students live in Morrison Hall and assume at
first that 50 of them love Chinese food and the
other 50 dont. If a person loves it, the
probability that he order it is 80 for each
meal. If the person doesnt, the probability is
10. Now, we observe an order of Chinese food has
been delivered to the hall. Given this event,
what is the probability the food was ordered by
someone who loves Chinese food?
36
Bayes Rule Let S1, S2, S3,…Sk represent the k
mutually exclusive, only possible states of
nature with prior probabilities P(S1),
P(S2),…P(Sk). If an event A occurs, the posterior
probability of Si given A is the conditional
probability
37
Bayes Rule
  • In English The denominator has all possible
    ways of getting there while the numerator has the
    one way you want
  • Chinese food example
  • P(LO) P(LO)/P(O)
  • P(O)P(OL)/(P(O)P(OL)P(O)P(OH))

38
  • Example 2
  • One part used in a production process is provided
    by three suppliers and supplier 1 provide 20, 2
    provide 30 and 3 provide 50. The percentage of
    defective parts supplied by these three suppliers
    are 0.05, 0.02, and 0.01, respectively. If a part
    randomly selected from the production process is
    found to be defective, what is the probability
    that the part is provided by supplier 1?

39
Lets Make a Deal
  • Monty Hall gives you the choice between 3 doors.
    One has a new car behind it and the other 2 have
    a goat. Once you have chosen a door Monty,
    knowing were the car is, reveals the contents of
    one of the doors you have not chosen (always the
    goat). He then gives you the option to choose
    one of the the other doors. Should you choose
    the other door?

40
  • Discrete Random variables and their probability
    distribution
  • Random variable a variable is called a random
    variable if the value that it assumes has a
    probability of occurring.
  • 1) Random variable is a quantitative variable
  • 2) We can use values from the outcome of an
    experiment to get random variables.

41
Examples of Random Variables
  • Toss a die
  • xnumber we observe
  • xnumber we observe-1
  • x0 if observe odd number 1 if
    observe even
  • Flip a coin
  • x 0 if H, x 1 if T
  • Select a student and measure their height
  • x height

42
Types of Random Variables
  • Discrete random variables have a countable
    number of values
  • Continuous random variables have infinite
    number of values
  • Examples
  • dice and coin toss discrete
  • height - continuous

43
Probability Distributions
  • Probability distribution for a discrete random
    variable is a formula, table, or graph that
    provides p(x), the probability associated with
    each of the value of x.
  • Example 1 toss a die, xnumber we observe
  • formula
  • table
  • graph
  • Example 2 toss two coins, what is the
    probability distribution of the the number of
    heads that we observe

44
  • Requirements for a discrete probability
    distribution
  • 0
  • Sum of P(x)1
  • Intuition each simple event is assumed one and
    only one value. And each value of x corresponding
    to one or more than one simple events, but
    different x cannot correspond to same simple
    event. So sum of P(x) is sum of simple event
    probability.

45
Expected Value
  • Expected value or population mean(mean) of random
    variable x with the probability distribution P(x)
    is given as
  • Where the elements are summed over all values of
    the random variable x.
  • Intuitionexpected value is weighted average
    value of x

46
Examples of Expected Values
  • If you throw a dice what is the expected value?
  • If you flip two coins what is the expected value?
  • Should you (if you have rational expectations)
    mail (37 cent stamp) a response to Publishers
    Clearing House for a chance to win 10,000,000
    when the odds of winning are one in 30,000,000?

47
Variance of a Random Variable
  • The variance of random variable x with
    probability distribution P(x) and expected value
    E(x) is given as
  • Where the summation is over all values of the
    random variable x
  • The Standard Deviation of random variable x is
    equal to the square root of its variance.

48
Example
  • A random variable has the following probability
    distribution
  • Construct a probability histogram
  • Find E(x), variance, and standard deviation
  • What does Tchebysheff tell us about this
    distribution

49
Example from article in class
  • A recent survey showed that 57 of French people
    have an unfavorable view of America. You ask at
    random 2 French folks what there opinion of
    America was, call x the number of French who have
    an unfavorable view.
  • Find the probability distribution and construct
    the histogram.
  • What is the probability that at least one will
    dislike American
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