Write

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Write

• What was the situation when we talked about the
  cereal boxes that contained athlete pictures?
• List as many details as you remember.
• When we addressed that situation using a
  simulation, how did we answer the question, How
  many boxes will we have to open until we get a
  Tiger Woods photo?

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Quiz Return

• Mostly really good!

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Finding Tiger
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Well, simulations are great, but

• The law of large numbers tells us that, as we
  simulate more, the value of the response variable
  will approach the true mean.
• Soooooo, what if we want to know that true mean?

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A word on Bernoulli

• Jakob Bernoulli (Basel, December 27, 1654 -
  August 16, 1705), also known as Jacob, Jacques or
  James Bernoulli was a Swiss mathematician and
  scientist and the older brother of Johann
  Bernoulli.
• He did not codify Bernoullis Principle, which is
  important his nephew Daniel did.
• He did work with Lebniz to shape up some of his
  early calculus.
• Hes one of my favorite mathematicians, but hes
  not pretty.

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Bernoulli Trials!

• A Bernoulli trial is just a particular type of
  situation, one which happens a lot
• there are exactly two possible outcomes
• success
• failure
• the probability of success (called p) is constant
• the trials are independent
• When were dealing with a situation like this,
  computing probabilities is pretty easy.

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Terms

• p probability of success
• q probability of failure
• of course, q 1-p

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The Geometric Model

• Consider the question,
• In a Bernoulli trial situation, what is the
  probability that we will have our first success
  on the Nth trial?
• We answer this question using what is called the
  geometric model.

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The Geometric Model

• The probability that the first success will occur
  on trial X is equal to
• P(X) qx-1p
• µ 1/p
• s ?(q/p2)

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So lets simulate first.

• Tiger is in 20 of boxes.
• Lets get random numbers, and use 1-20 to mean
  Tiger.
• Our response variable is the number of trials it
  takes to get a Tiger picture.
• Well each run 5 trials.

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Now, lets compute the number of boxes this
should take.

• p 0.2
• Find q.
• Find the expected value of X that is, the
  number of boxes it should take to find Tiger,
  using
• µ 1/p
• Then, find the standard deeeeev.
• s ?(q/p2)

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Practice Again.

• A basketball player makes 80 of his foul shots.
  Assuming independence (as usual), find the
  probability that in tonights game
• misses for the first time on his fifth attempt.
• makes his first basket on the fourth shot.
• makes his first basket on his first, second, or
  third shots.
• What is the expected number of shots it should
  take before he misses?

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Practice Again Again.

• Only 4 of people have type AB blood.
• How many people should we expect to have to check
  before we find one?
• Whats the probability that the first AB we find
  will be the 8th person?
• Whats the probability that we dont find an AB
  until the 40th person?

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Summary

• Bernoulli trials have three qualities
• There are two possible outcomes.
• The probability of success doesnt change.
• The trials are independent.
• A geometric model uses Bernoulli trials to
  estimate the number of trials before success.
• µ 1/p
• s ?(q/p2)

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Homework

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