Chapter 3Probability Sample Space and Sections 3.33.4

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Chapter 3Probability(Sample Space andSections
3.3-3.4)

• MATH320

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Sample Space
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Definition

• To compute probabilities
• Need to know the sample space
• Means all the events that could occur
• Example
• Coin Flip (head or tails)
• One coin H or T
• Two coins HH HT TH TT
• Die
• 1 2 3 4 5 6
• Multiple Choice
• A B C D E

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Tree Diagram

• Maps out all possible combinations
• Used for two or more experiments
• Start with the first outcome (experiment one)
• Add next outcome (experiment two or three)
• Connects to the previous branches
• Continue until all experiments/conditions have
  been branched

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Example Tree Diagrams
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Section 3.3Addition Rule
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Compound Event

• Any event combining two or more simple events.
• Notation
• P(A or B)
• P(event A occurs
• or event B occurs
• or they both occur)
• P(A)1/6
• P(B)1/6
• P(A or B)P(A)P(B)1/61/62/61/3

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General Rule for a Compound Event

• When finding the probability that event A occurs
  or event B occurs
• Find the total number of ways A can occur and the
  number of ways B can occur
• But find the total in such a way that no outcome
  is counted more than once.

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Compound Event

• Formal Addition Rule
• P(A or B) P(A) P(B) P(A and B)
• where P(A and B) denotes the probability that A
  and B both occur at the same time as an outcome
  in a trial or procedure. Not same as the
  multiplication rule.
• Intuitive Addition Rule
• To find P(A or B), find the sum of the number of
  ways event A can occur and the number of ways
  event B can occur, adding in such a way that
  every outcome is counted only once.
• P(A or B) is equal to that sum, divided by the
  total number of outcomes. In the sample space.

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Disjointed

• Events A and B are disjoint (or mutually
  exclusive) if they cannot both occur together.

Not Disjointed
Disjointed
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Example
Titanic Passengers

• Find the probability of randomly selecting a man
  or a boy.

Adapted from Exercises 9 thru 12
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Example
Men Women Boys Girls
Totals Survived 332 318 29 27
706 Died 1360 104 35 18
1517 Total 1692 422 64 56
2223

• Find the probability of randomly selecting a man
  or a boy.
• P(man or boy) 1692 64 1756 0.790
• 2223 2223 2223

Disjointed
Adapted from Exercises 9 thru 12
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Example
Men Women Boys Girls
Totals Survived 332 318 29 27
706 Died 1360 104 35 18
1517 Total 1692 422 64 45
2223

• Find the probability of randomly selecting a man
  or
• someone who survived.

Adapted from Exercises 9 thru 12
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ExampleFormal Addition Rule
Men Women Boys Girls
Totals Survived 332 318 29 27
706 Died 1360 104 35 18
1517 Total 1692 422 64 45
2223

• Find the probability of randomly selecting a man
  or
• someone who survived.
• P(man or survivor) 1692 706 332
  2066
• 2223 2223 2223 2223

0.929
NOT Disjointed
Adapted from Exercises 9 thru 12
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ExampleIntuitive Addition Rule
Men Women Boys Girls
Totals Survived 332 318 29 27
706 Died 1360 104 35 18
1517 Total 1692 422 64 45
2223

• Find the probability of randomly selecting a man
  or
• someone who survived.
• P(man or survivor) 1692 374 2066
• 2223 2223 2223

0.929
NOT Disjoint
Adapted from Exercises 9 thru 12
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Complementary Events

• P(A) and P(A)
• are
• mutually exclusive

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Rules of Complementary Events
P(A) P(A) 1 1 P(A)
P(A) 1 P(A)
P(A)
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Venn Diagram for the Complement of Event A
Figure 3-7
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Recap

• Compound events
• Formal addition rule
• Intuitive addition rule
• Dont double count
• Disjoint Events
• Complementary events

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Addition Rule Problems
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Disjointed

• Disjointed or Not Disjointed?
• Rolling a die and getting a 6 or 3
• Drawing a card from a deck and getting a club or
  an ace
• Tossing a coin and getting a head or tail
• Tossing a coin and getting a head and rolling a
  die and getting an odd number
• Drawing a card from a deck and getting a king or
  a club
• Rolling a die and getting an even number and a 5
• Tossing two coins and getting two heads or two
  tails
• Rolling two dice and getting doubles or getting a
  sum of eight

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Addition Rule of Probability

• In a box there are 6 white marbles, 3 blue
  marbles, and 1 red marble. If a marble is
  selected at random, what is the probability that
  it is red or blue?
• When a single die is rolled, what is the
  probability of getting a 2, 3, or 5?
• A storeowner plans to have his annual Going Out
  of Business Sale. If each month has an equal
  chance of being selected, find the probability
  that the sale will be in a month that begins with
  the letter J or M

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Addition Rule of Probability

• At a high school with 300 students, 62 play
  football, 33 play baseball, and 14 play both
  sports. If a student is selected at random, find
  the probability that the student plays football
  or baseball.
• A card is selected from a deck. Find the
  probability that it is a face card or diamond.
• The probability that a family visits Safari Zoo
  is 0.65 and the probability that a family rides
  on the Mt. Pleasant Tourist Railroad is 0.55. The
  probability that a family does both is 0.43. Find
  the probability that the family does both.

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Tree Diagram

• A box contains a 1 bill, a 5 bill, and a 10
  bill. Two bills are selected in succession with
  replacement. Find the probability that the total
  amount of money selected is
• 6
• Greater than 10
• Less than 15

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Section 3.4Multiplication Rule
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P(A and B)

• Any event combining two or more simple events
• Its the probability that both events will occur
• Notation
• P(A and B)
• P(event A occurs in the first trial
• and event B occurs in the second trial)
• P(A)1/6
• P(B)1/6
• P(A and B)P(A) x P(B)1/6x1/61/36

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Tree Diagrams
A tree diagram is a picture of the possible
outcomes of a procedure, shown as line segments
emanating from one starting point. These
diagrams are helpful in counting the number of
possible outcomes if the number of possibilities
is not too large.

• Figure summarizes
• the possible outcomes
• for a true/false followed
• by a multiple choice question.
• Note that there are 10 possible combinations.

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Example
Genetics Experiment Mendels famous
hybridization experiments involved peas, like
those shown in the figure (below). If two of the
peas shown in the figure are randomly selected
without replacement, find the probability that
the first selection has a green pod and the
second has a yellow pod.
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Example -- Solution
First selection P(green pod) 8/14 (14 peas,
8 of which have green pods)
Second selection P(yellow pod) 6/13 (13
peas remaining, 6 of which have yellow pods)
With P(first pea with green pod) 8/14 and
P(second pea with yellow pod) 6/13, we have
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Independent Events

• Two events A and B are independent
• If the occurrence of one does not affect the
  probability of the occurrence of the other.
• Several events are similarly independent if the
  occurrence of any does not affect the occurrence
  of the others.
• Playing California and New York Lotteries

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Dependent Events

• Occurrence of the first changes the probability
  of the second.
• Card selected from a deck. The probability of the
  second card pulled would be effected
• Probability of pulling an ace the first time and
  then a second time. No replacement
• P(first ace)4/52
• P(second ace)3/51
• P(two aces)4/52 3/510.00452 or 0.425
• Having your car start and getting to class on
  time.

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Independent or Dependent?

• Tossing a coin and selecting a card from a deck
• Driving on ice and having an accident
• Drawing a ball from an urn, not replacing it, and
  then drawing a second ball
• Having a high IQ and having a large hat size
• Tossing one coin and then tossing a second coin
• Dependent 2 3
• Independent 1, 4, 5

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Formal Multiplication Rule

• P(A and B) P(A) P(B A)
• Note that if A and B are independent events
• P(B A) is really the same as
• P(B)
• P(BA) means
• Probability of B
• Given that A has occurred

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Intuitive Multiplication Rule

• When finding the probability
• that event A occurs in one trial and B occurs in
  the next trial,
• multiply the probability of event A by the
  probability of event B,
• but be sure that the probability of event B takes
  into account the previous occurrence of event A.

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Example Important Principal

• Probability of pulling an ace the first time and
  then a second time. No replacement
• P(first ace)4/52
• P(second ace)3/51
• P(two aces)4/52 3/510.00452 or 0.425
• Example illustrates the important principle that
  the probability for the second event B should
  take into account the fact that the first event A
  has already occurred.
• P(B A) represents the probability of event B
  occurring after it is assumed that event A has
  already occurred (read B A as B given A.)

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Small Samples from Large Populations

• If a sample size is no more than 5 of the size
  of the population,
• treat the selections as being independent
• (even if the selections are made without
  replacement, so they are technically dependent).

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Summary of Fundamentals

• Addition Rule
• In the addition rule, the word or on P(A or B)
  suggests addition. Add P(A) and P(B), being
  careful to add in such a way that every outcome
  is counted only once.
• Multiplication Rule
• In the multiplication rule, the word and in P(A
  and B) suggests multiplication. Multiply P(A)
  and P(B), but be sure that the probability of
  event B takes into account the previous
  occurrence of event A.

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Recap

• Notation for P(A and B)
• Tree diagrams
• Notation for conditional probability
• Independent events
• Formal and intuitive multiplication rules

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Multiplication Rule Problems
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With and without replacements

• Two cards are drawn with replacements. Find the
  probability of getting two queens
• P(Q and Q) P(Q) x P(Q)
• 4/52 4/52 1/169 0.005917
• Two cards are drawn without replacements. Find
  the probability of getting two queens
• P(Q and Q) P(Q) x P(Q)
• 4/52 3/51 1/221 .004524

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Multiplication Rule of Probability

• In a study, there are 8 guinea pigs 5 are black
  and 3 are white. If 2 pigs are selected without
  replacement, find the probability that both are
  white.
• In a classroom there are 8 freshmen and 6
  sophomores. If three students are selected at
  random for a class project, find the probability
  that all 3 are freshmen.
• A large flashlight has 6 batteries. Three are
  dead. If two batteries are selected at random and
  tested, find the probability that both are dead.

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Rolling of the Dice Results
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Individual Events
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640 Events
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One I Did
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Summary
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Probability of Stuff
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Mega Millions

• Concept
• Match five balls
• Plus a power ball
• Combinations of five balls
• P(matching five balls)1/3,819,8160.00000026179
• P(matching power ball)1/460.2173913043
• P(both)0.00000000569

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Lotto 47

• Concept match 6 balls out of 47 total balls
• Combination of 6 balls
• P(matching 6 balls)0.00000009313