Binomial and Geometric Distributions

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Binomial and Geometric Distributions
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Properties of Binomial Experiments

• It consists of a fixed number of observations
  called trials.
• Each trial can result in one of only two mutually
  exclusive outcomes success (S) and failure (F)
• Outcomes of different trials are independent.
• The probability that a trial results in S is the
  same for each trial.

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Binomial Random Variable

• The binomial random variable x is defined as
• x number of successes observed when an
  experiment is performed
• The probability distribution of x is called the
  binomial probability distribution.

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Example

• A few days ago we studied the purchasing
  characteristics of customers shopping for a hot
  tub. We considered x number among four
  customers who selected an electric (as opposed to
  gas) hot tub.
• This is a binomial experiment with
• number of trials 4 P(success) P(E)0.4

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New Example

• Lets look at the case of five customers, a
  binominal experiment with five trials.
• Possible x values are 0, 1, 2, 3, 4, 5.
• How many possible outcomes are there? (Remember
  the fundamental counting principle?
• There are 32 possible outcomes, and five of them
  yield x1.
• SFFFF FSFFF FFSFF FFFSF FFFFS

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Example (cont)

• Lets take a look at the probability for the
  outcomes x1 (SFFFF, FSFFF, FFSFF, )
• The calculation will be the same for any outcome
  with only one success (x1).

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Example (cont)
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Example (cont)

• What about for x2. How many ways can we select
  two from among the five trials to be the Ss?
• What is the probability of P(SSFFF)?

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Example (cont)

• What is p(2) if we know
• Remember

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The Binomial Distribution

• Ready for this
• Let
• n number of independent trials in a binomial
  experiment
• p constant probability that any particular
  trial results in a success

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Example (Revisited)

• Lets go back to our 5 customers and look at p(2)
  using our new formula.
• We said.
• Does that fit with our formula?
• n 5 x2 p.4

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Next Example

• 60 of all watches sold by a large discount store
  have a digital display and the rest have analog.
  The type of watch purchased by the next 12
  customers will be noted.
• x number of watches that have a digital
  display.

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Next Example

• What is the probability that 4 watches are
  digital? Go ahead and try it now

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Using Appendix Table X

• To find p(x) for any particular value of x,
• Locate the part of the table corresponding to
  your value of n (5, 10, 15, 20, or 25)
• Move down to the row labeled with your value of
  x.
• Go across to the column headed by the specified
  value of p
• Try it now lets do n20 and p 0.8
• Find p(15).

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Homework

• 7.42
• 7.43 (cant use table)
• 7.44 (cant use table)
• 7.45 (use table)
• 7.47
• 7.48
• 7.49
• 7.50