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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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write
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?”
quiz return
Quiz Return
  • Mostly really good!
well simulations are great but
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?
a word on bernoulli
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 Bernoulli’s Principle, which is important – his nephew Daniel did.
  • He did work with Lebniz to shape up some of his early calculus.
  • He’s one of my favorite mathematicians, but he’s not pretty.
bernoulli trials
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 we’re dealing with a situation like this, computing probabilities is pretty easy.
terms
Terms

p = probability of success

q = probability of failure

  • of course, q = 1-p
the geometric model
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.
the geometric model9
The Geometric Model
  • The probability that the first success will occur on trial X is equal to

P(X) = qx-1p

    • μ = 1/p
    • σ = (q/p2)
so let s simulate first
So let’s simulate first.
  • Tiger is in 20 % of boxes.
  • Let’s 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.
    • We’ll each run 5 trials.
now let s compute the number of boxes this should take
Now, let’s 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.
    • σ = (q/p2)
practice again
Practice Again.
  • A basketball player makes 80 % of his foul shots. Assuming independence (as usual), find the probability that in tonight’s 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?
practice again again
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?
    • What’s the probability that the first AB we find will be the 8th person?
    • What’s the probability that we don’t find an AB until the 40th person?
summary
Summary
  • Bernoulli trials have three qualities:
    • There are two possible outcomes.
    • The probability of success doesn’t change.
    • The trials are independent.
  • A geometric model uses Bernoulli trials to estimate the number of trials before success.
    • μ = 1/p
    • σ = (q/p2)
homework
Homework

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