School Improvement in Maryland
Designing Models to Simulate Actual Events
 
Data Analysis and Probability .
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Objectives
 
  • The student will design models to simulate actual events using various random devices.
  • The student will estimate the likelihood of a particular outcome using results of simulation.
     
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Core Learning Goals
 
  • 3.1.1 :The student will design and/or conduct an investigation that uses statistical methods to analyze data and communicate results.
  • 3.1.3: The student will calculate theoretical probability or use simulations or statistical inferences from data to estimate the probability of an event.
     
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Drill
 
  • Consider a well-shuffled standard deck of 52 cards.

    1. What is the probability of drawing an Ace?
      P(Ace)=4/52=1/3
    2. What is the probability of drawing the Jack of Spades?
      P(Jack of Spades)=1/52
    3. What is the probability of drawing a face card (a Jack, Queen, King, or Ace)?
      P(face card)=16/52=4/13
    4. What is the probability of drawing a red card?
      P(red card)=26/52=1/2
       
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materials
 
  • Activities: "Guessing on a True/False Test," "Modeling with Other Random Devices," and "Simulating Families with Three Children"
  • Answer Keys
  • Coins, dice, spinners, or other similar devices
  • .  Print Version: (Acrobat 31k)
    The print version contains all student worksheets and answer keys needed to complete the lesson.
     

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Calculator Skills
 
    NONE
     
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activities
 
  1. Drill. The drill provides a review of calculating theoretical probabilities.
  2. Exploration. The first activity asks the question, "What is the probability of passing a test, for which you have not studied, by guessing?" This is certainly a question of interest to students. This activity is different from previous activities because students are no longer trying to make predictions about a population. It is known that one has a 50% chance of responding correctly to a true-false statement. It is known that there are ten statements on the test. What we don't know is the likelihood of responding correctly to at least 6 of these statements, thus passing the test. We can use a coin to model the correctness of each response, since there is a 50% chance of landing on either side. If we flip the coin ten times, we can model the results of one test. However, if we flip the coin another ten times (we call each set of ten flips a trial), we may get different results. So we flip the coin many, many times to estimate various frequencies in the long-run. The more trials we conduct, the closer our estimates will be to the theoretical probability. An explanation of the context of saves and attempted shots on goal may be required when completing the activity "Modeling with Other Random Devices".
  3. Class Discussion. It is often difficult to calculate theoretical probabilities, so we continue conducting trials (this is similar to sampling) in an attempt to estimate these values. Theoretical probabilities deal with long-run behavior, and because results vary we need many trials/samples to model this behavior. It is often difficult and impractical to conduct experiments or surveys over and over again, so we rely on simulation techniques to model natural phenomenon quickly.
  4. Additional problems. "Simulating Families with Three Children"
     
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Assessment
 
  • Summary Questions: A goalie saves 2/5 of the attempted shots on goal. Suppose there were eight attempted shots in a game.
    1. Can you use a coin to simulate the attempted shots in one game? Use mathematics to justify your answer.
      (No. There are not enough outcomes in coin tossing to model the shots on goal.)
    2. Can you use a six-sided number cube to simulate the attempted shots in one game? Use mathematics to justify your answer.
      (Yes. For example, let 1 and 2 represent making a shot, let 3, 4, and 5 represent missing the shot, and toss the cube again if it lands on 6.)
    3. Can you use this spinner to simulate the attempted shots in a game? Use mathematics to justify your answer. Insert spinner pic here (No. There are not enough outcomes on the spinner to model the shots on goal.)
    4. Can you use a standard deck of 52 cards to simulate the attempted shots in one game? Use mathematics to justify your answer. (Yes. For example, let all 10's, Jacks, Queens, Kings, and Aces represent making a shot, ignore all red 2's, and let all other cards represent missing the shot. Obviously, using a deck of cards would not be very efficient.)
       
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Homework
 
  • "Simulating Families with Three Children"
     
Objective Objective Core Learning Goals Core Learning Goals Drill Drill Materials Materials Calculator Skills Calculator Skills Activities Activities Assessment Assessment Homework Homework