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# Write - PowerPoint PPT Presentation

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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Presentation Transcript
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
• Mostly really good!
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…
• 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!
• 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

p = probability of success

q = probability of failure

• of course, q = 1-p
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 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.
• 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.
• 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.
• 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.
• 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
• 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

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