Chapter 8 Counting Principles: Further Probability Topics

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Chapter 8Counting Principles Further
Probability Topics

• Section 8.4
• Binomial Probability

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• Many probability problems are concerned with
  experiments in which an event is repeated many
  times.
• Probability problems of this kind are called
  Bernoulli trials, or Bernoulli processes.
• In each case, some outcome is designated a
  success, and any other outcome is considered a
  failure.
• Bernoulli trials problems are sometimes referred
  to as binomial experiments.

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Binomial Experiments

• The following criteria must be met
• 1.) The same experiment is repeated several
  times.
• 2.) There are only two possible outcomes,
  success and failure.
• 3.) The repeated trials are independent, so that
  the probability of success remains the same
  for each trial.

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• A single die is rolled four times in a row.
  Getting a 5 is a success, while getting any
  other number is a failure.
• a.) Why is this a binomial experiment?
• Already classified outcomes as successes or
  failures.
• Experiment is repeated several times.
• Outcomes are independent when rolling a single
  die.

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• A single die is rolled four times in a row.
  Getting a 5 is a success, while getting any
  other number is a failure.
• b.) Find the probability of having 3 successes
  followed by 1 failure.
• P(S ? S ? S ? F)

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• A single die is rolled four times in a row.
  Getting a 5 is a success, while getting any
  other number is a failure.
• c.) Find the probability of having 3 successes
  
• and 1 failure, in any order.
• Possible Outcomes
• SSSF SSFS SFSS FSSS
• Thus, P(3S ? 1F) 4 ( )

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Binomial Probability Formula

• If p is the probability of success in a single
  trial of a binomial experiment, the probability
  of x successes and n-x failures in n independent
  repeated trials of the experiment is
• n of trials
• x of successes
• p probability of success

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• Example A single die is rolled 10 times. Find
  the probability of rolling exactly 7 fives.
• n trials 10
• success rolling a five
• x successes 7
• p prob. of success 1/6

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• Example A restaurant manager estimates the
  probability that a newly hired waiter will still
  be working at the restaurant six months later is
  only 60. For the five new waiters just hired,
  what is the probability that at least four of
  them will still be working at the restaurant in
  six months? Assume that the waiters decide
  independently of each other.
• n 5
• x 4 or 5
• p 0.6

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• Example A flu vaccine has a probability of 80
  of preventing a person who is inoculated from
  getting the flu. A county health office
  inoculates 87 people. Find the probabilities of
  the following.
• a.) Exactly 15 of the people inoculated get the
  flu.
• b.) No more than 3 of the people inoculated get
  the flu.
• c.) None of the people inoculated get the flu.