Chapter 12: Probability

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Chapter 12 Probability

• Basic Concepts
• Events Involving Not and Or
• Conditional Probability Events Involving And
• Binomial Probability
• Expected Value

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Chapter 12 Probability

• 3. Conditional Probability Events Involving
  And

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Conditional Probability

• Define The probability of event B, computed on
  the assumption that event A has happened, is
  called the conditional probability of B given A.
  It is denoted as

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Example Conditional Probability
A red and green die are rolled. Event A is that
the sum of the dice equals 6. Event B is that
the same number is rolled on each die.
Find the conditional probability that the same
number is rolled on each die given that the sum
of the dice equals 6.
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Example Conditional Probability
The sample space is given on page 673 and also on
the next slide.
6
Example Conditional Probability
7
Example Conditional Probability

• Event Asum of dice is 6
• has cardinality 5
• Event same number is rolled on each die given
  that the sum of the dice equals 6
• has cardinality 1.

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Example Conditional Probability

• Therefore

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Conditional Probability

• Notice from the last example that the sample
  space is reduced from S with
• to
• where

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Conditional Probability Formula
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Previous Example Conditional Probability

• Event Asum of dice is 6
• Event B same number is rolled on each die
• Intersection

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Previous Example Conditional Probability

• Sample space has 36 members
• Therefore,

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Conditional Probability

• Warning
• and

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Previous Example Conditional Probability

• Therefore
• and

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Independent Events

• Define Two events A and B are called independent
  events if the occurrence of one of them has no
  effect on the occurrence of the other one.

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Independent Events

• Define Two events A and B are independent events
  means
• or

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Independent Events

• Example of independent events
• Roll a pair of dice and the events are Aeven on
  the first and Bodd on the second

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Independent Events

• 2. Example of events that are not independent
• Roll a pair of dice and the events are Aeven on
  the first and Bsum of dice is a four

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Multiplication Rule of Probability (Events
Involving And)

• If A and B are any two events, then

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Proof Multiplication Rule of Probability (Events
Involving And)

• Multiply both sides by P(A)
• to get the formula

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Example Multiplication Rule of Probability
(Events Involving And)

• Two marbles are drawn without replacement from a
  bag containing two green, three yellow, and four
  red marbles. Find the probability that you will
  draw
• Two green marbles
• No yellow marbles

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Example Multiplication Rule of Probability
(Events Involving And)

• Answers
• Two green marbles 1/36
• No yellow marbles 5/12

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Multiplication Rule of Probability (Events
Involving And)

• If A and B are any two independent events, then

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Proof Multiplication Rule of Probability (Events
Involving And)

• Independent events means that
• to get

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Example Multiplication Rule of Probability
(Events Involving And)

• Two marbles are drawn with replacement from a bag
  containing two green, three yellow, and four red
  marbles. Find the probability that you will
  draw
• Two green marbles
• No yellow marbles

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Example Multiplication Rule of Probability
(Events Involving And)

• Answers
• Two green marbles 4/81
• No yellow marbles 4/9

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Example Multiplication Rule of Probability
(Events Involving And)

• Not Independent Page 753, problem 26 (see page
  698 for card problems)
• Independent Page 754, problem 41,42,43 (note
  that 43 is the complement of problem 42!)
• Not Independent Page 754, problem 46
• Independent Page 755, problems 71,72,73

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Example Genetics

• Genetics of Plant Height
• Consider a population of plants that cannot
  self-pollinate
• The plant is diploid and has two copies of each
  gene one from the ovule (egg) and one from the
  pollen.
• Height is determined by a single gene which has
  two alleles (variants)
• b and B

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Example Genetics

• The B allele is dominant
• Plants with at least one copy of B are tall and
  those with two copies of b are short.
• Suppose that the parent plants are both with
  genotype Bb (one short allele and one tall
  allele)

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Example Genetics

• What is the probability that the next generation
  will be short?

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Example Genetics

• Answer
• Sample space of genotypes for next generation
• (see example on page 729)

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Example Genetics

• Answer
• Aevent next generation is short
• Probability next generation is short

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Example Genetics

• Suppose we are given a tall offspring, but we do
  not know its genotype. What is the probability
  that its genotype is BB?

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Example Genetics

• Answer
• This is an example of conditional probability
• Tevent that plant is tall
• Gevent that genotype is BB
• Find

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Example Genetics

• Answer
• Tevent that plant is tall
• Gevent that genotype is BB

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Example Genetics

• Answer
• and

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Example Colorblindness

• The probability of colorblindness depends on a
  persons sex.
• Mevent a person is male
• Fevent that person is female
• Cevent that person is colorblind
• Assume the following

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Example Colorblindness

• Find the probability that a male is colorblind.
• also
• Find the probability that a colorblind person is
  a male.

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Example Colorblindness

• Answer
• Find the probability that a male is colorblind.
• 5 of all males are colorblind

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Example Colorblindness

• Answer
• Find the probability that a colorblind person is
  a male.
• Need to determine

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Example Colorblindness

• Answer
• The event that you are colorblind
• Colorblind and Male OR Colorblind and Female
• Note
• and
• are mutually exclusive (do not intersect)

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Example Colorblindness

• Answer

43
Example Colorblindness

• Answer
• Find the probability that a colorblind person is
  a male.
• 83 of colorblind people are male