Probability Using Dice

Transcription

1 Using Dice One Page Overview By Robert B. Brown, The Ohio State University Topics: Levels:, Statistics Grades 5 8 Problem: What are the probabilities of rolling various sums with two dice? How can you analyze this situation? Getting Started: Ask what sums are possible if two dice are rolled. Do all the different sums that you came up with have the same chance of happening? Is it easier to think about these questions if the two dice are of different colors or of the same color, or does it even matter? 1. Number, Number Sense and Ohio Academic Content Standards, 2002 NCTM Principles and Standards, x 1. Number, Number Sense and 1. Number, Number Sense and 1. Number and x 1. Number and 2. Measurement 2. Measurement 2. Measurement 2. Algebra 2. Algebra 3. Geometry and Spatial Sense 4. Patterns, Functions and Algebra Mathematical Processes Reasoning X 3. Geometry and Spatial Sense 4. Patterns, Functions and Algebra Mathematical Processes Reasoning X 3. Geometry and Spatial Sense 4. Patterns, Functions and Algebra 5. Data Analysis and Mathematical Processes 3. Geometry 3. Geometry 4. Measurement 4. Measurement X 6. Problem Solving 6. Problem Solving 7. Reasoning and Proof x 7. Reasoning and Proof 8. Communication 8. Communication 9. Connections 9. Connections 10. Representation 10. Representation Note: Capital X denotes major emphasis; lower case x denotes minor emphasis. Using Dice p. 1

2 Using Dice By Robert B. Brown, The Ohio State University Topics:, Statistics Levels: Grades 5 8 Materials: A pair of dice of the same color for each pupil Timing: One hour Prerequisites: Knowledge of fractions Problem: What are the probabilities of rolling various sums with two dice? How can you analyze this situation? Goals: Introduce the notion of a set of outcomes and the probability of a particular outcome Provide a theoretical analysis of dice rolling using a simple example which can be used later to illustrate many simple notions in probability and statistics Experience the difference between a set of outcomes that are not equally likely and a set which are equally likely Big Ideas: Equally likely Statistics Using Dice p. 2

3 Procedure: 1. Give each pupil a pair of dice of the same color. Ask them what sums are possible when they roll the dice. After a little bit of thought they will agree that all sums from 2 to 12 and no others are possible. 2. Ask them if they think that all of these sums have the same chance of being rolled? They will probably agree that the answer is no. They may reason that 2 and 12 have the smallest chance of being rolled because there is only one way to get a 2 and only one way to get a 12, but other sums can be gotten in more than one way. 3. Tell them that they are now going to analyze the chances that any particular sum is rolled. To help with this give each pupil a pair of dice of different colors, let s just say that the colors are red and green. 4. Have everyone take their red die. Ask what could they get if they roll the red die by itself? The numbers 1 6, of course. Are the chances the same for each of these? They will agree the answer is: yes, if the die is a good one. Is it the same for the green die by itself? Sure. 5. If you roll the two dice together, the red one and the green one, and you write down what comes up on each of them, how many possibilities are there? Some will say at first 12 possibilities, 6 for the red and 6 for the green. Have them do some actual rolls and write down all the different outcomes. In short order they will see that the number of different outcomes is 36, which is 6 6, rather than Ask them if they think that all of the 36 different outcomes have the same chance of occurring? (This is a good opportunity to introduce the terminology equally likely outcomes if you want to.) They will probably agree that all 36 outcomes are equally likely. 7. Now you are ready to calculate the probabilities of the different sums. Ask for the probability of getting a 2. Of the 36 equally likely outcomes only one yields a sum of 2, namely, red 1 & green 1. So the chance of getting a sum of 2 is 1 out of 36. What about a sum of 3? That you can get in two ways, red 1 & green 2 and red 2 & green 1. So the chances of getting a sum of 3 are 2 out of 36. This is an opportunity to introduce more terminology if you want to. You can explain that the technical way of expressing the chances of getting a 3 is, The probability of getting a sum of 3 is 2/36. When you do that, they may observe that 2/36 is the same as 1/18. But writing it as 1/18 obscures the fact that these probabilities were calculated using 36 equally likely outcomes, and it makes more difficult the comparison of the probabilities of different sums. Using Dice p. 3

4 Extensions: Tie probability to statistics by comparing theoretical results with experimental results. It is quick and easy to collect data by rolling dice. If you break the students into small groups and let each group present data on the sum of two dice for, say, 36 rolls, they will see that rolling only 36 times can miss the predicted distribution of outcomes by quite a bit due to randomness alone. In fact, the data from different groups of pupils is likely to miss the predicted distribution in different ways. Then you can have the students pool their data to see how much closer the data get to the predicted distribution with a larger number of trials. The Mathematics: Replacing two dice of the same color with two dice of different colors makes it easier to count the 36 different equally likely outcomes for each roll. Sum Number of ways (red 1 & green 2 and red 2 & green 1) As a check, add up the column number of ways, and you will get 36. To see that these probabilities for sums do not change when the differently colored dice are replaced by dice of the same color, you can ask the pupils to imagine that all of the color on each die gets concentrated at one tiny dot on each die. This concentration does not change the probabilities of the different sums, but now it looks like they are just rolling two dice of the same color. If you give them dice of the same color to work with, the following situation may come up. They will analyze all the different ways of getting sums with dice of the same color and come up with the following: Using Dice p. 4

5 Sum Number of ways & & & 3, 2 & & 4, 2 & & 5, 2 & 4, 3 & & 6, 2 & 5, 3 & & 6, 3 & 5, 4 & & 6, 4 & & 6, 5 & & & 6 If you add up the number of ways you get 21. Therefore one might think that the probabilities have denominator 21. It might take some experimenting to see that these 21 legitimately counted outcomes are not equally likely. For example, look at the sum of 4. Using a red and a green die there is only one way to get 2 & 2. But you can get 1 & 3 in two ways, red 1 & green 3 and red 3 & green 1. So getting 1 & 3 is twice as likely as getting 2 & 2. Using two dice of the same color, it would become apparent after a lot of rolls that 1 & 3 is more likely than 2 & 2. Using Dice p. 5

6 Relationships to the Ohio Academic Content Standards, 2002: Grades 5 7: Data Analysis and Standard Find all possible outcomes of simple experiments or problem situations, using methods such as lists, arrays and tree diagrams. Describe the probability of an event using ratios, including fractional notation. Make and justify predictions based on experimental and theoretical probabilities. Mathematical Processes Standard Use inductive thinking to generalize a pattern of observations for particular cases, make conjectures, and provide supporting arguments for conjectures. Grades 8 10: Data Analysis and Standard Compute probabilities of compound events, independent events, and simple dependent events. Make predictions based on theoretical probabilities and experimental results. Mathematical Processes Standard Apply reasoning processes and skills to construct logical verifications or counter examples to test conjectures and to justify and defend algorithms and solutions. Relationships to the NCTM Principles and Standards, 2000: Grades 6 8: Data Analysis and Standard Instructional programs from pre kindergarten through grade 12 should enable all students to Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. Understand and apply basic concepts of probability. Reasoning and Proof Standard Instructional programs from pre kindergarten through grade 12 should enable all students to Make and investigate mathematical conjectures. Using Dice p. 6