### Overview

Investigate the probability of the sums of two six - sided die

### Background information

Probability can be determined in one of two ways: theoretical and experimental. It is very rare that students are able to understand the probability of the outcomes of sums 2 - 12 without charting all possible values.

### Concepts

1. The probability of a certain sum of two die is equal to the total number different sum combinations for each possible sum out of the total number of all possible sums.
2. Each die has six sides.
3. Each die is the same.
4. Each side has an equal chance of being rolled.
5. Each die is fair.
6. The sums of the die are: 2, 3, 4, 5, 6, 7, 8, 9, 110, 11, 12.
7. There is one way for the dice to have a sum of 2, two for 3...six for 7 five for 8... For a total of 36.
8. The probability of a specific outcome is equal to the total number of ways to achieve the same outcome out of all the total number of different outcomes possible.
9. The probability of an outcome is the number of specific outcomes out of the total number of all possible outcomes of one event.
10. Theoretical probability is found by generating all the possible outcomes or combinations of events.
11. Theoretical probability doesn't usually match the experimental probability.
12. Theory is an idea used to explain or predict an event.
13. Experiment is a test made to try to understand something.

### Misconceptions

I can cause a certain number to be rolled (blowing, throw hard, throw a certain way...). It is magic.

### Assessment

Have students explain how to find the experimental probability and theoretical probability of each of the following. Sums of dice with the number of sides different than six. Sums of dice with two different number of sides. Spinners with unequal partitions and or different colors of sections and the probability of getting pairs of colors. Create problems with different amounts of different colors of socks in a drawer and pulling out one sock at a time and what kinds of pairs would be made that way.

Bloom’s Taxonomy If students have never experienced the concept and derive the concept on their own it would be application or possibly synthesis. If they have conceptualized the concept before it is comprehension.

### Objective

Students predict the outcome of the sum of two dice rolled 36 times, roll the dice 36 times and record the sums, chart the data, draw a conclusion about probability and communicate the difference between theoretical and experimental probability and how to find each.

### Materials

Dice, pencil, paper

### Beginning

Ask the student to predict what the outcome would be if they tossed two six sided dice at a time, summed the dice, recorded it, and repeated the process 36 times. Ask them how they determined their prediction. Record all suggestions on the board. Have the students roll two dice 36 times, record the numbers and the sums for each roll.

### Middle

1. Ask students how to display data. If students do not know how to arrange data have them chart the number of rolls for each sum 2 - 12 (2 - 12 horizontal axis, # rolls vertical axis). Data could also include the addends for each sum.
2. Students put their data on the board.
3. Ask questions like the following to see how they interpret the data.
4. What pattern do you see from the data?
5. What sum turned up most?
6. What are the odds of each sum turning up?
7. Analyze the data by having students explain the pattern. It may be necessary to list every pair of addends for each sum 2 - 12 theoretical probability.
8. Have students communicate the pattern and compare the experimental probability with the theoretical probability