Lesson+Notes

By: Emily Bolon, Marc Aisenbrey, Kaitlin Oetken, and Darla Brandt A.  Chance has no memory B.  Probability can be characterized along a continuum from impossible to certain C.  The probability of an event, which is the chance that a given event will occur, is between 0 and 1 D.  The larger the number of trials, the better the estimate will be   E.   __Theoretical probability:__ exact probability determined by analysis of the event F.  __Simulation:__ model designed to have the same probabilities as the real situation A.  Likely or Not Likely? 1.  Focus on possible or impossible Examples: a.  Create table of “possible” and “impossible” using nursery rhymes b.  Ask students to judge and come up with certain, impossible, or possible events 2.  Help children develop chance or probability on a continuum of more or less likely Examples: a.  __Race to the Top:__ Students use spinners of varying partitions to race colors b.  Have students connect frequency charts to spinners and provide explanations c.  __Add, Then Tally__: Students roll two dice, add sum, and record on tally sheet A.  Introduce the idea of a continuum from impossible to certain Examples: 1.  Have students organize spinners of varying proportions on a continuum 2.  __Design a Bag__: Allow students to determine color of items in a bag for certain percent positions 3.  __Testing Bag Designs__: Have students test bag designs created in previous activity by drawing items from a bag ten times; make large bar graph or tally graph of class data
 * Probability Notes **
 * Chapter 22: Exploring Concepts of Probability **
 * I.  **** Big Ideas of Probability  **
 * II.  **** Introducing Probability  **
 * III.  **** The Probability Continuum **

A.  __Probability:__ measure of the chance of an event; measure of certainty of an event Types: 1.  Likelihood of occurrence of event known a.  Determined by examining all possibilities b.  Example: Fair die has a 1/6 chance of producing each number 2.  Likeliness of occurrence of event is not observable a.  Determined through __empirical data__, which is evidence from past experiments or data collection b.  This type is most applicable to most fields that use probability c.  Example: Basketball player’s likelihood of making a free throw based on previous record B.  Theoretical Probability 1.  When all possible outcomes of a simple experiment are equally likely a.  Number of out comes in the event/ Number of possible outcomes 2.  When answer requires empirical data b.  Number of observed occurrences of the event/ Total number of trials 3.  Problem-based Method of introducing theoretical probability a.  __Fair or Unfair?-__ Engage students in an unfair game and have them determine if it is theoretically fair __ Example games for students to analyze- __ *Rolling two coins with points given to one player for two heads, another player for two tails, and another player for a heads and a tails *Rock, paper, scissors in normal way or adapted so “same” scores 1 point and “different” scores 1 point C.  Experiments 1.  Used for probabilities that cannot be determined by the theoretical likeliness or the theoretical probability 2.  Probability for these events can be determined through empirical evidence, which may be preexisting or established through experimentation 3.  __Law of Large Numbers:__ phenomenon that the relative frequency becomes a closer approximation of the actual probability or the theoretical probability as the size of the data set (sample) increases *In other words, this means the larger the size of the data set, the more representative that sample is of the population. Example Activity: a.  __Cup Toss-__ Have students predict and test possible landings of a cup; pool data and approximate actual probability of different landings 4.  __ Misconception of many students__- “law of small numbers” or belief that probability should play out in the short term Example activity for students to overcome misconception: a.  __Get All 6!-__ Have students roll a die ten times and mark each number that is rolled on a frequency table; have students compare frequency charts in each case; focus on the fact that data varies in the short term and evens out in the long term b.  __What are the chances__?- Have students explore combining trial spins and viewing the changes that occur as trials increase; try with other experiments such as Cup Toss 1.  Develops an appreciation for a simulation approach to solving problems, which is often done to solve real-world problems 2.  Significantly more intuitive- results begin to make sense and are not as abstract 3.  Eliminates guessing at probabilities and wondering about correctness 4.  Provides an experiential background for examining the theoretical model 5.  Helps students see how the ratio of a particular outcome to the total number of trials begins to converge to a fixed number- For an infinite number of trials, the relative frequency and theoretical probability would be the same 6.  It is a lot more fun and interesting! 1.  __Advantages__ (if students accept the result generated by the technology as truly random or equivalent to the hands-on device) a.  Makes content more accessible b.  Enables more trials in less time __ Examples __ *Electronic devices, including some simple calculators and graphing calculators, are designed to produce random outcomes with a press of a button *Computer software is available that flips coins, spins spinners, or draws numbers from a hat *A variety of tools can show graphical data of trials 2. Website to explore: National Library of Virtual Manipulatives a. Has outstanding virtual experiments 1.  __Sample space__: set of all possible outcomes for that experiment *Example: Bag with 2 red, 3 yellow, & 5 blue would have a sample space of 10 tiles 2. __Event__: subset of the sample space *Example: Event of drawing a yellow tile has 3 elements or outcomes in the sample space 3. __One-event experiment__: requires one activity to determine an outcome *Examples: rolling a single die or drawing one colored chip from a bag 4. __Two-event experiment__: requires two or more activities to determine an outcome *Examples: rolling two dice or drawing two cubes from a bag 5. __Independent Events:__ occurrence or nonoccurrence of one event that has no effect on the other *Example: Rolling dice- result on one die does not effect the other a. Common error: failure to distinguish between the two events, especially when results are combined b. To create the sample space for two independent events, it is helpful to use a chart or diagram that keeps the two separate and illustrates all possibilities 6. Solving Two-event Probabilities a. One way to determine the theoretical probability is to list all outcomes and count the number of outcomes that make up the event *__Limitations:__ events may not all be equally likely and can be very tedious b. Another way is to create an __Area Model.__ *__Advantages__**:** --accessible to a range of learners --less abstract than equations or tree diagrams --reasonable to perform more than two independent events --use of and and or connectives can be modeled effectively --clear to students, without memorization of formulas, how to find probabilities of independent events 7. Dependent Events: occur when the second event depends on the result of the first a. Example: There are two identical boxes. One has a two counterfeit dollar bills and the other has one of each. You must choose one box and from that box select one dollar. *Probability of getting a dollar in second event depends on which box is chosen first 1.  Definition- technique used for answering real-world questions or making decisions in complex situations where an element of chance is involved 2.  Why Simulations? a.  Can be too dangerous, complex, or expensive to manipulate real situation * Example: In designing a rocket, a large number of related systems all have some chance of failure. Various combinations of failures can cause serious problems. Knowing the probability of serious failures will help determine if backup systems are required. * Solution: A model that simulates all of the chance situations can be designed and run repeatedly with the help of a computer; they can estimate chance of failure 3.  Useful Guide of Steps for any Simulation a.  Identify key components and assumptions of the problem b.  Select random device for the key components- any random device that has outcomes with the same probability as the key component can be selected c.  Define a trial, which consists of simulating a series of key components until the situation has been completely modeled one time d.  Conduct a large number of trials and record the information e.  Use the date to draw conclusions __Key Idea:__ Translation of real-world information into models is the essence of applied mathematics 4.  Ways simulations can be used to gather empirical data a.  Simple calculators can generate random numbers that can stand for heads and tails, boys and girls, true or false, and any other pair of equally likely outcomes. __*Remember__- Students who use random number generators will need some direction to use them to their advantage. b.  In graphing calculators, random numbers can be produced and stored in a list. The list can then be displayed graphically. A TI-83 will “roll” as many dice as you request. A histogram will display the totals for each sum. Using the TRACE feature, the value for each bar can be displayed. c.  TI-73 calculator has a built in dice and coin function.
 * IV.  **** Theoretical Probability and Experiments  **
 * V.  **** Implications for Instruction (Reasons for an experimental approach to probability is important to use in middle grades classroom) **
 * VI.  **** Use of Technology in Experiments **
 * VII.  **** Sample Spaces and Probability of Two Events **
 * VIII.  **** Simulations  **