Simulating Experiments

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Simulating Experiments
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• A couple plans to have children
• until they have a girl or until
  they
• have 4 children.
• What are the chances they will
• have a girl?

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How can we answer that question?

• Actually have a lot couples carry out the
  experiment and calculate the relative frequency
  of having a girl.

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• 2. Develop a probability model and use it to
  calculate a theoretical answer.
• (COMING ATTRACTION)

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• 3. Start with a model that in some way reflects
  the experiment and then simulate it a number of
  times

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Simulation Steps

• State the problem or describe the experiment.
• What are the chances that a certain major league
  baseball player will have a streak of 4 hits in
  his next 20 at-bats?

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• State the assumptions
• At bats are independent of each other
• The player will continue to get a hit 1/3 of the
  time he is at bat

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• Assign digits to represent outcomes
• Random number table digits 0-32 represent a
  hit 33-99 represent no hit

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• Simulate many repetitions
• Looking at 20 consecutive two-digit entries in
  the table represents one simulation.
• Keep track of whether there was a 4-hit streak
  in each simulation.

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• State your conclusions
• Estimate the probability by using a proportion
• Number of trials with desired outcome
• Total number of trials

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CAUTIONS

• 25 Repetitions is not enough to be confident in
  our conclusion.
• Computers can do thousands of simulations quickly
  and cheaply and give us more reliable results.

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Question

• How would you use the following aids to establish
  correspondence in a simulation that involves a
  75 chance?

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A coin
Toss the coin twice. HH, HT, TH, TT
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A Six-sided Die
1, 2, 3, 4. Discard 5, 6
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A Random Digit Table
Two digits 00 74 75 99
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A Standard Deck of Playing Cards
Club, Spade, Heart / Diamond
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Randomizing with a Calculator

• MATH
• PRB
• 5randInt( ENTER
• randInt(lowest, highest, of repetitions)
• TI-89 CATALOG F3 (Flash Apps)

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Nuclear Safety

• A nuclear reactor is equipped with two
  independent automatic shutdown systems to shut
  down the reactor when the core temperature
  reaches the danger level.

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• Neither system is perfect. System A shuts down
  the reactor 90 of the time when the danger level
  is reached.
• System B does so 80 of the time.
• The reactor is shut down if either system works.

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• How would you simulate the response of System A
  to a dangerous temperature level?

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• How would you simulate the response of System B
  to a dangerous temperature level?

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• Both systems are in operation simultaneously.
  Combine your previous answers to simulate the
  response of both systems to a dangerous
  temperature level.

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Demand for Cheesecake

• The owner of a bakery knows that the daily demand
  for a highly perishable cheesecake is as follows

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• /Day Relative Frequency
• 1 0.05
• 2 0.15
• 3 0.25
• 4 0.20
• 5 0.10

In other words, 1 cheesecake is sold 5 of the
time, 2 cheesecakes are sold 15 of the time, 3
cheesecakes are sold 25 of the time, 4
cheesecakes are sold 20 of the time, 5
cheesecakes are sold 10 of the time. Zero
cheesecakes are sold what percentage of the time?
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• Use simulation to find the demand for the
  cheesecake on 30 consecutive business days.

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• Suppose that it cost the baker 5 to produce a
  cheesecake, and that the unused cheesecakes must
  be discarded at the end of the business day.

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• Suppose also that the selling price of a
  cheesecake is 13.
• Use simulation to estimate the number of
  cheesecakes that he should produce each day in
  order to maximize profit.

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• /Day Relative Frequency
• 0 0.25
• 1 0.05
• 2 0.15
• 3 0.25
• 4 0.20
• 5 0.10

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Correspondence

• 0 24 0 cakes demanded
• 25 29 1 cake demanded
• 30 44 2 cakes demanded
• 45 - 69 3 cakes demanded
• 70 - 89 4 cakes demanded
• 90 99 5 cakes demanded

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Random Generator

• 10 53 75 10 92 56 42 38
• 3 0 3 4 0 5 3 2 2
• 98 16 61 82 40 57 32 57
• 3 5 0 3 4 2 3 2 3
• 77 64 62 4 12 29 34 33
• 0 4 3 3 0 0 1 2 2
• 4 26
• 4 0 1

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• How many cheesecakes would you make every day?