Probability Models Target Goals: I can define and list a sample space for an event. I can use basic probability rules.

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Probability ModelsTarget Goals I can define
and list a sample space for an event.I can use
basic probability rules.
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• Weve done models for lines and for distributions
  of data (normal density curves).
• Now we are going to give a mathematical
  description or model for randomness.

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Sample Space

• The set of all possible outcomes of an event is
  the sample space S of the event.
• Example For the event roll a die and
• observe what number it lands on. The
• sample space contains all possible
• numbers the die could land on.
• S 1, 2, 3, 4, 5, 6

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An Event

• An event is an outcome (or a set of outcomes)
  from a sample space.
• Example 1 When flipping three coins,
• an event may be getting the result
• HTH.
• In this case, the event is one
• outcome from the sample space.

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• Example 2 When flipping three coins,
• an event may be getting two tails.
• In this case, the event is a set of outcomes
  (HTT, TTH, THT) from the sample space.
• An event is usually denoted by a capital letter.
• For example, call getting two tails
• The probability of event A is denoted

event A.
P(A).
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• Example Roll the Dice
• Give a probability model for the chance process
  of rolling two fair, six-sided dice one thats
  red and one thats green.

• Probability Rules

Since the dice are fair, each outcome is equally
likely. Each outcome has probability 1/36.
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Roll a 5 is an event that we will call A.

• List the possible outcomes for A
• A (1, 4),(2, 3),(3, 2),(4,1)
• What is the P(A)?
• P(A) 4/36 or 11.11

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• To specify S, we must state what constitutes an
  individual outcome and then state which outcomes
  can occur.
• You must be able to list the outcomes in a
  sample space.

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Listing Outcomes

• Example Random Digit
• If you let your pencil fall on the Table of
  random digits, what are the possible outcomes S?
• S 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

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Real Life

• In statistics, what might tossing a coin
  represent?
• An opinion poll with a yes or no answer.
• Any two outcome event.

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Probability Rules

• These facts follow from the idea of probability
  as the long-run proportion of repetitions on
  which an event occurs.

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Rule 1

• The probability of any event is between 0 and 1.
• Probability of 0 indicates the event will never
  occur.
• Probability of 1 indicates the event will always
  occur.
• 0 P (A) 1

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Rule 2

• If S is the sample space, then P(S) 1.
• All possible outcomes in the sample space add up
  to 1.

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Rule 3

• The probability that event A does not occur is
  one minus the probability that A does occur.
• Called the complement of A
• Is denoted Ac.
• P(Ac ) 1 P(A)

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These complementary events can be shown on a Venn
Diagram. E 2, 4, 6 and EC 1, 3, 5
Let the circle represent event E.
Let the rectangle represent the sample space.
Let the shaded area represent event not E.
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• Example
• When flipping two coins, the probability of
  getting two heads is
• (.5)(.5) 0.25.
• The probability of not getting two heads is

1 0.25 0.75.
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Rule 4

• If events A and B are disjoint then,
• They have no outcomes in common.
• The events never occur simultaneously.
• The probability that A or B occurs is the
  probability that A occurs plus the probability
  that B occurs.
• P(A or B) P(A) P(B)

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A Venn Diagram for the roll of a six-sided die
and the following two events A 2 B
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A and B are mutually exclusive (disjoint) since
they have no outcomes in common
The intersection of A and B is empty!
3
4
6
2
1
5
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Example

• Let event A be rolling a die and landing on an
  even number, and
• event B be rolling a die and landing on an odd
  number.
• The outcomes for A are 2, 4, 6 and the outcomes
  for B are 1, 3, 5.
• These events are disjoint because they have no
  outcomes in common.

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• So the probability of A or B (landing on either
  an even or an odd number) equals the probability
  of A plus the probability of B.
• P(A) P(B) ½ ½ 1

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Disjoint Events
(mutually exclusive)

• Ways to express
• A U B s/a A or B, or
• A n B empty set

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Venn diagram

• Venn diagram showing disjoint (mutually
  exclusive) events A and B.

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Complement

• A U Ac S
• A n Ac empty set
• Venn diagram showing the complement Ac of an
  event A.

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Example Marital Status of Young Women

• Draw a woman age 25 to 34 years old at random
  and record her marital status.
• What does at random mean?
• Every woman has the same chance to be chosen.

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Here is the probability model
Marital status Never married Married Widowed Divorced
Probability 0.298 0.622 0.005 0.075

• Why do the probabilities add up to 1?
• P(S) 1

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• Find the probability the woman drawn is not
  married by the complement rule.
• P(not married) 1 P(married)
• P(not married) 1 0.622 0.378

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Are never married and divorced are disjoint
events?

• Yes, because a woman cant be both never
  married and divorced at the same time.
• P(never married or divorced)
• P(never married) P(divorced)
• 0.298 0.075 0.373

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Example Benfords Law

• Fake numbers in tax returns and other settings
  often display patterns that arent present in
  legitimate records.
• The first digits in legitimate records often
  follow a distribution called Benfords Law.

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Benfords probability of first digits. Note
0 not a leading digit here.
First digit 1 2 3 4 5 6 7 8 9
Probability .301 .176 .125 .097 .079 .067 .058 .051 .046

• a. Events A first digit 1,
• B first digit is 6 or greater
• P(A)
• P(B)

P(1) .301
P(6) P(7) P(8) P(9)
0.222
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Describe in words P(Ac).

• A first digit 1
• First digit anything but a 1
• P(Ac) 1 - P(A)
• 1 - 0.301
• 0.699

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b. Are events A and B disjoint?

• A first digit 1,
• B first digit is 6 or greater
• Yes, they cant occur together so
• P(A or B) P(A) P(B)
• 0.301 0.222
• 0.523

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c. P(C) first odd digit

• P(1) P(3) P(5) P(7) P(9)
• 0.609

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d. Is P(B or C) disjoint?

• B first digit is 6 or greater
• P(C) first odd digit
• No, we cant add the probabilities.
• Outcomes 7 and 9 are common to both events so,
• P(1) P(3) P(5) P(6) P(7) P(8) P(9)
• 0.7227

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Equally Likely Outcomes

• Most random phenomena do not have equally likely
  outcomes.
• When they do, the probability of any event A is
• P(A) count of outcomes in A
• count of outcomes in S

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Find the P(B) using equally likely outcomes.

• Event B first digit is 6 or greater
• P(B) P(6) P(7) P(8) P(9)
• 1/9 1/9 1/9 1/9 4/9
• 0.444
• Summary
• A crook who tries to use random digits will end
  up with too many first digits 6 or greater.

vs. Benfords 0.222