12.5 Binomial Distribution

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12.5 Binomial Distribution

• Objectives
• Find the probability of r successes in n trials
  of
• a binomial experiment.

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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 (what youre
  looking for) or failures (what youre not looking
  for).
• The trials are identical and independent with a
  constant probability of success, p, and a
  constant probability of failure, 1-p.

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Binomial Probability

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

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Example 1
Suppose that the probability a seed will
germinate is 85. What is the probability that
exactly 7 of the seeds will germinate when 10 are
planted.
n r p 1-p
10 7 0.85 0.15
P 0.13 or 13
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Binomial Probability

• In a binomial experiment, the events r successes
  and s success are mutually exclusive so you can
  find the probability of exactly r successes or s
  successes by adding the probabilities.
• P(r successes or s successes)
• P(r successes) P(s successes)

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Example 2
The probability of an ambulance being operational
is 0.96. Find the probability that at least 6 of
8 ambulances are operational.
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).
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Practice

• A person randomly selects answers to all 10
  questions on a multiple-choice quiz. Each
  question has 1 correct answer and 3 incorrect
  answers. What is the probability that the
  student gets exactly 7 correct?
• A batter has a batting average of 0.250. What is
  the probability that the batter will get at least
  3 hits in the next 5 at bats?