Conditional Probability

1
Conditional Probability

• Conditional Probability Defined
• The Multiplication Rule
• Partitions
• The Law of Total Probability
• Independence of Events

2
Materials for Review and Practice

• Student Notebook
• Slides 106 thru 127 (pp. 70 81, Conditional
  Probability)
• Slides 128 thru 140 (pp. 81 87, Partitions)
• Student Manual
• pp. 67 - 76
• Conditional Probability Worksheet

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Conditional Probability

• We begin with a simple example
• Toss a balanced die once and record the number on
  the top face. (What is the associated probability
  model?)
• Let E be the event that a 1 shows on the top
  face.
• Let F be the event that the number on the top
  face is odd.
• What is P(E)?
• What is P(E) if we are told that the number on
  the top face is odd, that is, we know that the
  event F has occurred?

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Conditional Probability

• Key idea The original sample space no longer
  applies. The new (or reduced) sample space is
• SF1, 3, 5
• Notice that the new sample space consists only of
  the outcomes in F.
• P(E occurs given that F occurred) 1/3
• Notation P(EF) 1/3

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Conditional Probability

• Definition. If A and B are events and P(B)?0,
  then we define the conditional probability of A
  given B by

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Conditional Probability Graphically
Reduced Sample Space
Original Sample Space
7
Conditional Probability

• Conditional probability problems lend themselves
  to several methods of solution. We can
• Apply the definition
• Construct Venn Diagrams
• Construct Tree Diagrams
• Use Contingency Tables

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Example One

• The probability that event A occurs is .63.
• The probability that event B occurs is .45.
• The probability that both events occur is .10.
• Find the following probabilities by first using
  the definition and then using Venn diagrams.
• P(AB)
• P(BA)
• P(ABC)
• P(ACB)
• P(ACBC)

The following identities may prove useful For
any two events A and B
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Example Two

• The table below shows the results of a survey of
  1100 traffic accidents. The data is tabulated
  according to the cause of the accident and the
  sex of the driver at fault.
• Given that an accident was caused by mechanical
  failure, what is the probability that the driver
  was male?
• Given that the driver was a male, what is the
  probability that the accident was caused by
  mechanical failure?

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Example Three

• If at least one child in a family with two
  children is a girl, what is the probability that
  both children are girls?
• Let G be event child is a girl and B be event
  child is a boy. What is the known condition? At
  least one child is a girl so S GG, GB, BG

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Conditional Probability

• Multiplication Rule
• Partitions
• Law of Total Probability

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Multiplication Rule

• Using the definition of conditional probability
• we multiplying both sides of the equation by P(B)
  to get
• This is known as the multiplication rule of
  probabilities and provides another way at
  computing the joint probability of A and B.

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Example Four

• A publisher estimates that among the books they
  publish 52 have become best sellers and that
  about 78 of all books they publish get good
  reviews in The New York Times. Furthermore, if
  the probability that a book which has become a
  best seller gets a good review in the New York
  Times is 0.87, what is the probability that a
  book which got a good review in the New York
  Times will become a best seller?

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Multiplication Rule and Trees
Conditional Probability
Basic Probability
Probability
Event
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Partitions

• Suppose that the nonempty sets (events) E and F
  have the following two properties
• (1)
• (2)
• or equivalently
• (1 )
• (2 )
• Then E and F are said to partition S.

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Partitions
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Law of Total Probability
Let the events E and F partition the finite
discrete sample space S for an experiment and let
A be an event defined on S.
Start
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Example Five

• Your retail business is considering holding a
  sidewalk sale promotion next Saturday. Past
  experience indicates that the probability of a
  successful sale is 60, if it does not rain.
  This drops to 30 if it does rain on Saturday. A
  phone call to the weather bureau finds an
  estimated probability of 20 for rain.
• What is the probability that you have a
  successful sale?

Source Thomson, R.B. and Lamoureux, C.G.
(2003). Mathematics for Business Decisions, Part
1 Probability and Simulation. MAA Washington,
DC
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Example Five Continued

• Let R be the event that it rains next Saturday
  and let N be the event that it does not rain next
  Saturday.
• Let A be the event that your sale is successful
  and let U be the event that your sale is
  unsuccessful.
• We are given that P(AN) 0.6 and P(AR) 0.3.
  
• The weather forecast states that P(R) 0.2.
• Our goal is to compute P(A).

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Example Five Continued
Bayes, Partitions
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Example Five Continued

• P(A) P(R ? A) P(N ? A)
• 0.06 0.48
• 0.54

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Example Six

• Suppose that we have two identical jars labeled I
  and II. Jar I contains 4 red marbles and 3 blue
  marbles. Jar II contains 5 red marbles and 1
  blue marble. Select a jar at random and then
  select 1 marble from the jar and note its color.
  A typical question we might ask about this
  process is
• What is the probability that the ball drawn is
  blue?

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Example Six and a half

• Suppose 2 balls are drawn from a box in
  succession without replacement. The box contains
  5 red balls and 8 green balls.
• Draw a tree diagram representing this situation.
• Find probability second is red given that the
  first was red.

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Example Seven

• Three manufacturing plants A, B, and C supply 20,
  30 and 50, respectively of all shock absorbers
  used by a certain automobile manufacturer.
  Records show that the percentage of defective
  items produced by A, B and C is 3, 2 and 1,
  respectively.
• What is the probability that a randomly chosen
  shock absorber installed by the manufacturer will
  be defective?
• Consider the following questions
• What is the probability that the part came from
  manufacturer A, given that the part was
  defective?
• What is the probability that the part came from
  manufacturer B, given that the part was not
  defective?
• To answer these questions will require Bayes
  Theorem.

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Conditional Probability

• Independent Events

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Independent Events

• If the probability of the occurrence of event A
  is the same regardless of whether or not an event
  B occurs, then the events A and B are said to be
  independent of one another. Symbolically, if
• then A and B are independent events.

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Independent Events

• Recall that
• Then we can state the following relationship for
  independent events
• if and only if A and B are independent events.

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Example Eight

• A fair coin is tossed twice. Find the
  probability that
• the second toss is a head
• the second toss is a head given that the first
  toss is a head.

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Example Nine

• A card is drawn from a standard deck of 52 cards
  and then a second card is drawn without replacing
  the first card.
• Let A be the event that a second card is a spade
  and let B be the event that the first card is a
  spade.
• Determine whether A and B are independent events.

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Example Ten

• Toss two fair die, one green and one red and
  observe the numbers on the top faces. Decide
  which pairs of events, A and B, are independent
• 1. A the sum is 5 B the red die shows a
  2
• 2. A the sum is 5 B the sum is less than
  4
• 3. A the sum is even B the red die is
  even

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Example Eleven

• Two cards are drawn from a well-shuffled deck of
  52 cards. Find the probability that they are
  both aces if the first card is
• Replaced
• Not replaced

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Independence

• The notion of independent events can be extended
  to conditional probabilities. If E, F, and G are
  three events, then E and F are independent, given
  that G has happened, if

Likewise, events E, F, G are independent, given
that an event H has happened, if
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Independence vs. Mutually Exclusive

• Are they the same? NO
• (ex) main computer fails back up fails. They
  are independent but not mutually exclusive
• (ex) Two students who do not know each other are
  in the same class. They are independent but not
  mutually exclusive. If they know each other,
  events could be dependent and not mutually
  exclusive.

34
Project

• What do conditional probabilities add to solving
  out project?
• When we computed E(X) in the last section we used
  P(S) .464 and P(F) .536 but these numbers
  failed to take into account the information on
  the borrower.
• Let Y be the event borrower has 7 years experience

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Project

• Let T be the event that the borrower has a
  Bachelors Degree.
• Let C be the event that the economy is normal.
• We want P(SYnTnC) which we cannot find since the
  database does not give this information.

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Project

• But we can make an assumption which is logical
  and that is that our events, Y, T and C are
  independent.
• This assumption will allow us to use Bayes
  Theorem to solve our problem.
• We will look at this in the next section

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Project what we can do now

• P(SY) P(SBRYBR) where SBR means these are the
  borrowers at BR bank where the attempted workout
  was successful and YBR are all the borrowers at
  BR bank with 7 years experience.
• P(ST) P(SCJTCJ) CJ means Cajun
• P(SC) P(SDUCDU) DU means Dupont

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Project

• P(SBRYBR) 105/239 .439
• P(FBRYBR) 134/239 .561
• P(SCJTCJ) 510/1154 .442
• P(FCJTCJ) 644/1154 558
• P(SDUCDU) 807/1547 .522
• P(FDUCDU) 740/1547 .478
• Why do these numbers add up to 1?

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Project

• Let ZY ZT and ZC be the random variables that
  represents the amount of money Acadia Bank
  receives from workouts with clients with 7 years
  experience, Bachelors Degrees and in normal
  economic times.
• E(ZY) 4,000,000(.439) 250,000(.561)
  1,896,250. Similarly, E(ZT) 1,907,500 and
  E(ZC) 2,207,500

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Project

• Recall what we need is P(SYnTnC) which we still
  cannot do.
• But P(YnTnCS) P(YS)P(TS)P(CS) P(YBRSBR
  )P(TCJSCJ)P(CDUSDU) because Y, T and C are
  independent events.
• P(YnTnCS) 0.022
• P(YnTnCF) 0.021