BINOMIAL DISTRIBUTIONS

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BINOMIAL DISTRIBUTIONS

• A medical center has 8 ambulances. Given the
  ambulances current condition, regular
  maintenance, and restocking of medical supplies,
  the probability of an ambulance being operational
  is 0.96. Find the probability that at least 6 of
  the 8 ambulances are operational. We will solve
  this problem later.

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BINOMIAL EXPERIMENT

• A probability experiment is a binomial experiment
  if both of the following conditions are met
• The experiment consists of n trials whose
  outcomes are either successes (the outcome is the
  event in question) or failures (the outcome is
  not event in question.)

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BINOMIAL EXPERIMENT (CONT)

• In a binomial experiment, the trials are
  identical and independent with a constant
  probability of success, known as p, and a
  constant probability of failure, known as 1 p.
  In other words, since there is a constant
  probability of success, this means that the
  probability of success will be the same for each
  and every trial of the experiment.

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Finding a Binomial Probability

• In a binomial experiment consisting of n trials,
  the probability, P, of r successes (where 0 r
  n , p is the probability of success, and 1-p is
  the probability of failure) is given by the
  following formula
• P nCr pr (1 p)n-r

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Example Finding a Binomial Probability

• According to a survey taken by USA TODAY, about
  37 of adults believe that UFOs really exist.
  Suppose you randomly survey 6 adults. What is the
  probability that exactly 2 of them believe that
• UFOs really exists?

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SOLUTION

• Let p 0.37 be the probability that a randomly
  selected adult believes that UFOs really exist.
  By survey 6 adults, you are conducting n 6
  independent trials. The probability of getting
  exactly r 2 successes is
• P(r2) 6C2 (0.37)2 (1- 0.37)6-2
• (0.37)2(0.63)4
• 0.323, or 32

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Constructing a Binomial Distribution
Probability
Number of Believers in UFOs
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SOLUTION TO UFO PROBLEM

• The previous slide showed the binomial
  distribution in a histogram.
• Here are the solution values for the individual r
  values
• P(r 0) 6C0(0.37)0(0.63)6 0.063
• P(r 1) 6C1(0.37)1(0.63)5 0.220
• P(r 2) 6C2(0.37)2(0.63)4 0.323
• P(r 3) 6C3(0.37)3(0.63)3 0.253
• P(r 4) 6C4(0.37)4(0.63)2 0.112
• P(r 5) 6C5(0.37)5(0.63)1 0.026
• P(r 6) 6C6(0.37)6(0.63)0 0.003
• The probability of getting at most r 2
  successes is
• P(r 2) P(2) P(1) P(0) .323 .220
  .063 0.606
• The probability that at most 2 of the people
  believed that UFOs really exist is about 61

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Back to the

• Problem Find the probability that at least 6 of
  8 ambulances are operational. Round to the
  nearest tenth of a percent.
• At least 6 ambulances are operational when
  exactly 6,7,or 8 ambulances are operational.
• Find P(exactly 6) P(exactly 7) P(exactly 8)
• Use n 8, p 0.96, and 1- p 1 0.96
• Use the Binomial Probability formula that we used
  in the UFO problem. It is on slide 4.

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Chicken Problem

• For a science project you are incubating 12
  chicken eggs. The probability that a chick is
  female is 0.5.
• Start by calculating each binomial probability
  using this formula
• P (r) nCr (0.5)r (0.5)n-r nCr (0.5)n
• Then draw a histogram of the binomial
  distribution based on the probability that
  exactly r of the chicks are female.
• Finally, find the most likely number of female
  chicks.
• What do you notice about the distribution of bars
  on your histogram? How does it compare with the
  UFO histogram?