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CHAPTER 22 Exploring Concepts of Probability

CHAPTER 22 Exploring Concepts of Probability. Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville. Big Ideas. Chance has no memory.

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CHAPTER 22 Exploring Concepts of Probability

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  1. CHAPTER 22 Exploring Concepts of Probability Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville

  2. Big Ideas • Chance has no memory. • The probability that a future event will occur can be characterized along a continuum from impossible (0) to certain (1). • The probability of an event is a number between 0 and 1 that is a measure of the chance that a given event will occur. • The relative frequency of outcomes (of experiments) can be used as an estimate of the probability of an event. • For some events, the exact probability can be determined by an analysis of the event itself. • Simulation is a technique used for answering real-world questions or making decisions in complex situations in which an element of chance is involved.

  3. Probability • Notions of chance and fairness should begin early. • Students should have developed some intuition about how likely an outcome might be. • Initially focus on possible and not possible and move to impossible and certain. • Probability is a ratio that compares the desired outcome to the total possible outcomes.

  4. Introducing ProbabilityTry this oneActivity 22.2 Is it Likely? Ask students to judge events as certain, impossible, or possible • It will rain tomorrow. • Drop a rock in water and it will sink. • A sunflower seed planted today will bloom tomorrow. • A hurricane or tornado will hit our town. • You will have two birthdays this year. • You will be in bed before midnight. For each event students should justify their choice.

  5. Use of Random Devices • Random devices- spinners, number cubes, coins to toss, colored cubes drawn from a bag, color tiles etc. • Use random devices with which to count the outcomes.

  6. Probability Continuum Posting a probability continuum can be used as a reference for other opportunities to talk about how likely something is.

  7. Theoretical Probability and Experiments • Probability- two types • First type- when all possible outcome are equally likely • Second type- find out the probability by theoretically examining empirical data • Impossible to conduct infinite number of trials- consider relative frequency for large number of trials.

  8. Theoretically Probability Try this oneActivity 22. 8 Fair or Unfair? Rules • Play in a group of 3. • The game lasts for 20 tosses. • Toss 2 two-color counters (or coins e.g. pennies or 2 nickels). • Player A scores 1 point if both are red. • Player B scores 1 point if both are yellow. • Player C scores 1 point if there is one of each. Reflection Questions Is the game fair? Can you justify why or why not?

  9. Reflections on Fair or Unfair

  10. Experiments – existing data or sufficiently large number of trials

  11. Law of Large Numbers Phenomenon that the relative frequency of an event becomes closer to the actual probability or theoretical probability. • Larger the data set the more representative the sample is of the population • Comparing small data sets to large data sets guides the thinking about the importance of the size of trials.

  12. Why Use Experiments? • Helps students address common misconceptions • Model real-world problems that are actually solved by conducting experiments • Provide a connection to counting strategies (lists, trees, diagrams) • Provide an experimental background for examining the theoretical model • Help students see how the ratio of a particular outcome to the total number of trials begins to converge to a fixed number • Help students learn more than students who do not engage in experiments

  13. Independent Events • To create the sample space for two independent events use a chart or diagram • Examples- • rolling an even sum with two dice • Spinning blue twice on a spinner • Having a tack or cup land up when tossed once

  14. Dependent Events • Dependent events occur when the second event depends on the result of the first. • Example- game show to win a car if you make it through a maze to the room with the key. • Record with area model

  15. Simulations- technique for answering real-world problems

  16. Simulations • Identify key components and assumptions of the problem. • Select a random device for the key components. • Define a trial. • Conduct a large number of trials and record the information. • Use the data to draw conclusions.

  17. Addressing Misconceptions about Probability • Commutativity confusion- need to think of all the ways events can occur when trying to determine how likely an event is. • Gambler’s Fallacy- the notion that what has already happened influences the event. • Law of small numbers-expect small samples to be like the greater population • Possibility counting- see what the possibilities are and assume each is equally likely.

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