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

• 4. Binomial Probability

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Motivational Example

• Genetics
• In mice an allele A for agouti (gray-brown,
  grizzled fur) is dominant over the allele a,
  which determines a non-agouti color. Suppose
  each parent has the genotype Aa and 4 offspring
  are produced. What is the probability that
  exactly 3 of these have agouti fur?

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Motivational Example

• A single offspring has genotypes

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Motivational Example

• Agouti genotype is dominant
• Event that offspring is agouti
• Therefore

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Motivational Example

• Let G represent an agouti offspring and N
  represent non-agouti
• Exactly three agouti offspring may occur in four
  different ways (in order of birth)
• NGGG, GNGG, GGNG, GGGN
• These events are mutually exclusive!

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Motivational Example

• P(exactly 3 offspring has agouti fur)
• The four births are independent of each other so
• 
  etc.

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Motivational Example

• P(exactly 3 offspring has agouti fur)

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Motivational Example

• Is there a better way to compute this
  probability?

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Binomial Probability

• If the outcomes of an experiment consist of just
  2 categories, often described as
• Success or Failure,
• the associated probabilities are called
  binomial.

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Binomial Probability

• Examples
• Flip a coin
• Successheads
• Failuretails
• Mice produce offspring
• Successagouti fur,
• Failurenon-agouti fur

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Bernoulli Trials

• Repeated trials of an experiment which can be
  labeled as success or failure, where the
  probability of success is the same each time, are
  called Bernoulli trials.

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Bernoulli Trials

• Examples
• Flip a coin multiple times
• Mice produce many offspring

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Binomial Probability Formula

• n number of repeated trials
• p probability of success on any given
  trial
• q the probability of failure on any given
  trial (note q1-p)
• x number of successes that occur

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Binomial Probability Formula
Recall n factorial is
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Binomial Probability Formula Mice Example

• n 4 (number of repeated trials)
• p 3/4 (probability of success on
  any given trial)
• q 1/4 (probability of failure on any given
  trial)
• x 3 (number of successes that occur)

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Binomial Probability Formula
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Binomial Probability Formula

• Example page 761, problem 44
• At a large midwestern university, 80 of all
  students have their own personal computers. If
  five students at that university are selected at
  random, find the probability that exactly 3 of
  them have computers.

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Binomial Probability Formula Mice Example

• n 5 (number of repeated trials)
• p .80 (probability of success on
  any given trial)
• q .20 (probability of failure on any given
  trial)
• x 3 (number of successes that occur)

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Binomial Probability Formula