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Probability PowerPoint PPT Presentation

Probability Properties of probabilities 0 ≤ p(A) ≤ 1 0 = never happens 1 = always happens A priori definition p(A) = number of events classifiable as A total number of classifiable events A posteriori definition p(A) = number of times A occurred total number of occurrences So:

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Probability

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Probability l.jpg

Probability


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Properties of probabilities

  • 0 ≤ p(A) ≤ 1

    • 0 = never happens

    • 1 = always happens

    • A priori definition

  • p(A) = number of events classifiable as A

    total number of classifiable events

    • A posteriori definition

  • p(A) = number of times A occurred

    total number of occurrences


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So:

p(A)= nA/N = number of events belonging to subset A vs. the total possible (which includes A).

If 6 movies are playing at the theater and 5 are crappy but 1 is not so crappy what is the probability that I will be disappointed?

5/6 or p = .8333


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Probability in perspective

  • Analytic (classical) view

    • The common approach: if there are 5 bad movies and one good one I have an 83% chance in selecting a bad one.

    • Fisher

  • Relative Frequency view

    • Refers to the long run of events: the probability is the limit of chance i.e. in a hypothetical infinite number of movie weekends I will select a bad movie about 83% of the time

    • Neyman-Pearson

  • Subjective view

    • Probability is akin to a statement of belief and subjective e.g. I always seem to pick a good one.

    • Bayesian


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Experiment

  • Experiment -- a Process that produces outcomes

    • More than one possible outcome

    • Only one outcome per trial

  • Trial -- one repetition of the process

  • Event -- an outcome of an experiment

    • may be an elementary event, usually represented by an uppercase letter, e.g., A, E1


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  • Generally we can calculate the probability of one of a set of equally likely events by counting the sample space

  • Many problems in probability can be solved in this way

  • probability very often makes use of combinatorics (permutation and combination – we’ll talk about this later)


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Sample Space -- Set Notation for Random Sample of Two Families

  • S = {(x,y) | x }

    • x is the family selected on the first draw

    • y is the family selected on the second draw

  • Concise description of large sample spaces


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Sample Space

  • The set of all elementary events for an experiment

  • Methods for describing a sample space

    • Listing

    • Venn diagram


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Family

Children in Household

Number of Automobiles

Listing of Sample Space

(A,B), (A,C), (A,D),

(B,A), (B,C), (B,D),

(C,A), (C,B), (C,D),

(D,A), (D,B), (D,C)

A

B

C

D

Yes

Yes

No

Yes

3

2

1

2

Sample Space -- Listing Example

  • Experiment: randomly select, without replacement, two families from the residents of Denton

  • Each ordered pair in the sample space is an elementary event, for example -- (D,C)


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Y

X

Venn

Diagram

Venn Diagrams and the union of sets

  • Venn diagrams helps us pictorially represent many of the algebraic rules of probability

  • The union of two sets contains an instance of each element of the two sets.


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Y

X

Venn

Diagram

Intersection of Sets

  • The intersection of two sets contains only those elements common to the two sets.


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Definitions

  • Mutually exclusive events

    • both events cannot occur simultaneously.

      • Can’t be a junior and senior

    • Complementary events

      • Two mutually exclusive events that are all inclusive

  • Independent events:

    • occurrence of one event has no effect on the probability of occurrence of the other


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Y

X

Mutually Exclusive Events

  • Events with no common outcomes

  • Occurrence of one event precludes the occurrence of the other event

Venn

Diagram


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Complementary Events

  • All elementary events not in the event ‘A’ are in its complementary event.

Sample

Space

A


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  • Exhaustive sets

    • set includes all possible events

    • the sum of probabilities of all the events in the set = 1

  • Equal likelihood

    • roll a fair die each time the likelihood of 1-6 is the same whichever one we get, we could have just as easily have gotten another

    • Counter example- put the numbers 1-7 in a hat. What’s the probability of even vs. odd?


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AND vs. OR

  • How do we find the probability of one event or another occurring?

  • How do we find the probability of one event and another occurring?


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Addition

  • p(A or B) = p(A) + p(B)

    • Probability of getting a grape orlemon skittle in a bag of 60 pieces where there are 15 strawberry, 13 grape, 12 orange, 8 lemon, 12 lime?

    • p(G) = 13/60p(L) = 8/60

    • 13/60 + 8/60 = 21/60 = .35 or a 35% chance we’ll get one of those two flavors when we open the bag and pick one out


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Y

X

General Law of Addition

(not necessarily mutually exclusive)


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Married (Yes/No)

Yes

No

Total

Children (Yes/No)

.70

.14

.56

Yes

.19

.11

No

.30

.33

1.00

.67

Total

Example: Marriage and Children


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S

N

.56

.70

.67

General Law of Addition


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Married (Yes/No)

Yes

No

Total

Children (Yes/No)

.70

.14

.56

Yes

.19

.11

No

.30

.33

1.00

.67

Total

Contingency Table


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Multiplication

If A & B are independent

  • p(A and B) = p(A)p(B)

  • p(A and B and C) = p(A)p(B)p(C)

  • Probability of getting a grape and a lemon after two draws (with replacement) from the bag

    • p(Grape)*p(Lemon) = 13/60*8/60 = ~.0288


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Conditional probabilities

If events X and Y are not independent then:

  • p(X|Y) = probability that X happens given that Y happens

    • The probability of X “conditional on” or “given” Y occurs

    • It’s our ‘and’ type of question from before so we are going to use multiplication, however we don’t have independent events so it will be a little different

  • p(A and B) = p(A)*p(B|A)


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  • Example: once we grab one skittle we aren’t going to put it back (sampling without replacement) so:

    • p(A and B and C) = p(A)*p(B|A)*p(C|A,B)

    • Probability of getting grape and lemon on successive turns = p(G)*p(L|G)

    • (13/60)(8/59) = .0293


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Conditional and Joint Probabilities

  • A conditional probability is one where you are looking for the probability of some event with some sort of information in hand, e.g. the odds of having a boy given that you had a girl already.

  • A joint event or probability would be the probability of a combination of events e.g. that you have a boy and a girl for children

  • Aside: In this case the conditional would be higher b/c if we knew there was already a girl that means they’re of child-rearing age, probably interested in having kids etc. We have some additional info that would help us if we were just drawing out people randomly from some population.


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Conditional Probability

Venn Diagram

N

S

.56

.70

*note before with our previous conditional probability we were dealing with mutually exclusive events i.e. can’t be grape and lemon at same time


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Gender and Political Affiliation


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  • Let’s do a conditional probability: If I have a male, what is the probability of him being in the ‘Other’ category? Formally:

  • p(A|B) = p(A and B)/p(B) =

  • p(O|M) = p(O and M)/p(M) =

    = (.412*.714)/.588= .5


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  • Easier way by looking at table- there are 10 males and of those 10 (i.e. given that we are dealing with males) how many are “Other”?

  • p(O|M) = 5/10 or 50%.


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Joint Probability Example

  • What is the probability of obtaining a Female Independent from this sample?

  • In this case we’re looking for the joint probability of someone who is Female and Independent out of all possible outcomes:

    2/17 = 11.8


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Practice

  • a. What is the probability a person is over 25?

  • b. What is the probability that people under 25 spend at least 10 hours on the internet?

  • c. What is the probability that someone who does not spend 10 hours on the internet each week is over 25?

  • d. What is the probability of picking someone who spends less than 10 hours/wk on the internet and is under 25?


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  • a. 45/110 = .41

  • b. 50/65 = .77

  • c. 20/35 = .57

  • d. 15/110 = .14


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