Probability With Number Cubes

Today’s Learning Goals  We will continue to understand the link between part-whole ratios, decimals, and percents.  We will continue to develop an understanding of how to use counting trees to make an organized list of all the outcomes that could happen with different situations.  We will continue to develop an understanding of the link between experimental and theoretical probability.  We will continue to develop an understanding of determining when a game is fair or not.  We will continue to develop the ideas of equally likely and non-equally likely outcomes.

Review Previously, we used various strategies to find probabilities associated with games of chance. We found experimental probabilities by playing a game several times and evaluating the results. Also, we found theoretical probabilities by analyzing the possible outcomes of a game. What strategy did we use to list out all of the possible outcomes of a game? Great…we constructed a counting tree.

Again, we will use our graphing calculators to simulate rolling dice. First, press the “Apps” button to go to the applications screen. APP Rolling Dice Today, we will play a couple of games with dice (number cubes). Cursor down to highlight the Probability Simulation application and press “Enter”. Enter 46

Matching Colors Press any key, cursor down to select the “Roll Dice” option, and then press “Enter”. If we wanted to set up our dice a particular way, what button would we press to go to the setup screen? Enter Nice…the “Zoom” button corresponds to the set option. Make your setup screen look like the one at the right. Make sure you have two dice.

Matching Colors Now your calculator is set up to roll two six-sided dice at one time. Which button should you press to select OK? Good…the “Graph” button corresponds to OK. What button should you press to roll the dice? Awesome…the “Window” button corresponds to ROLL.

The Addition Game The first game we will play is called The Addition Game. Let’s play this one together as a whole class. I will be Player A and all of you students will be Player B. We will roll the two dice 36 times and keep score on the worksheet provided. If the sum of the numbers rolled is odd, I score 1 point. If the sum of the numbers rolled is even, you students score 1 point. The player with the most points after 36 rolls wins.

The Addition Game Using the students’ calculators, play the game and roll the dice 36 times. Based on the data of all 36 rolls, what is the experimental probability of rolling an odd sum? Based on the data, what is the experimental probability of rolling an even sum? Yes…

The Addition Game Let’s construct a counting tree to list all of the possible outcomes for the addition game. start st Die What were the possible outcomes on the first die? Yes…numbers 1 through 6.

The Addition Game If you got a 1 on the first die, what are the possible outcomes on the second die? start st Die 2 nd Die The same is true if you got a 2, 3, 4, 5, or 6 on the first die. Awesome…again numbers 1 through 6.

The Addition Game Based on the counting tree, how many total outcomes are there every time you roll the dice? start st Die 2 nd Die Yes…36 possible outcomes.

The Addition Game Based on the counting tree, how many total outcomes produce an odd sum? start st Die 2 nd Die Good…18 outcomes produce an odd sum.

The Addition Game Based on the counting tree, what’s the theoretical probability of getting an odd sum? start st Die 2 nd Die Great…½ OR 50% because 18 out of 36 are odd.

The Addition Game Based on the counting tree, how many total outcomes produce an even sum? start st Die 2 nd Die Good…18 outcomes produce an even sum.

The Addition Game Based on the counting tree, what’s the theoretical probability of getting an even sum? start st Die 2 nd Die Great…½ OR 50% because 18 out of 36 are even.

The Addition Game Do you think that The Addition Game was fair the way it was set up? Explain why or why not. Yes…because we both had an equal chance of getting points AND the point amounts were the same.

The Addition Game Min-wei wanted to give Player A 1 point if the sum was 2, 6 or 7 and Player B 1 point if the sum was any other number. Is her game fair? Explain. start st Die 2 nd Die No…because there is 12/36 ways for Player A to get 1 point and 24/36 ways for Player B to get 1 point.

The Addition Game Because in Min-Wei’s game, Player A has a 12/36  33% chance of getting a point, and Player B has a 24/36  66% chance of getting a point, what could we do to the assignment of points to make the game fair? Yes…because Player B has double the chances of getting points, we can let Player A have 2 points every time he has a chance to get points to even it out.

Partner Work You have 20 minutes to work on the following problems with your partner.

For those that finish early Royce invented a game based on the sum of two number cubes. In his game, Player A scores 3 points if the sum is a multiple of 3, and Player B scores 1 point if the sum is not a multiple of 3. Is Royce’s game a fair game? Explain why or why not.

Big Idea from Today’s Lesson The assigning of points can offset the difference in how often a player gets points to make it fair. For example, suppose that Player A has 3 times the amount of probability of getting points as Player B. Then, to make the game fair, Player B should get 3 times the amount of points as Player A when he has the chance to get points. Counting trees are very helpful for determining the possible outcomes when there are two or more things that occur.

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