# Discussion on Law of Large Numbers

b’Subject: Mathematics’
b’Topic: Dice (Law of Large Numbers) Project’
b”

Math 1001: Project – Law of Large Numbers                           Name:  _________________________

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Assigned Monday, October 26, 2020

In this project, you are going to be rolling two dice and adding the sum on the dice. We are going to explore the concept of the “Law of Large Numbers”. You can print this out and write on it or type right into this page.   Since this is a project (worth 20% of your grade), it needs to be done in a nice and neat format. I have put a copy of graph paper in the content area if you would like to have more room for your graphs. If I can’t read your writing, I will assume the answers are wrong.  It must be turned into the drop box by Friday, November 13, 2020 by 11:59 pm. There will be 10 points per day (including weekends) penalty for late projects.

Look back through your notes if you need to for definitions to see how to make a frequency distribution table, etc.

Directions:

1. Use two dice or the website https://www.random.org/dice/?num=2 (copy and paste into a browser) to roll 100 times. Go to Games (at the top), on the drop down menu, pick dice.  You can put in two dice to roll.

1. For each roll, record the SUM of the dice.
2. If the sum is even, put an “X” in the parity box. Otherwise, leave it blank.
3. For each group of 25, record the number of each of the sums you rolled. Make sure these numbers total 25 because it is easy to miscount.
4. On page 4, create a probability frequency distribution using the included table. These are broken into groups of 25. The first column will be for rolls #1-25 (only), the second will be for rolls #1-50 (only), the third will be for the rolls #1-75 (only), and the fourth will be for all of the rolls. (You are cumulating the results.) For the last column, if you cannot remember how to find the probability of rolling 2 dice, look back through your Section 7A notes and the table we created in class.
5. Answer the questions in complete sentences.

 Roll # Total Parity Roll # Total Parity 1 26 2 27 3 28 4 29 5 30 6 31 7 32 8 33 9 34 10 35 11 36 12 37 13 38 14 39 15 40 16 41 17 42 18 43 19 44 20 45 21 46 22 47 23 48 24 49 25 50

For rolls #1 – 25:                                                                        For rolls #26 – 50:

Number of 2s:  ______________                                              Number of 2s:  _______________

Number of 3s:  ______________                                              Number of 3s:  _______________

Number of 4s:  ______________                                              Number of 4s:  _______________

Number of 5s:  ______________                                              Number of 5s:  _______________

Number of 6s:  ______________                                              Number of 6s:  _______________

Number of 7s:  ______________                                              Number of 7s:  _______________

Number of 8s:  ______________                                              Number of 8s:  _______________

Number of 9s:  ______________                                              Number of 9s:  _______________

Number of 10s:  ______________                                            Number of 10s:  _______________

Number of 11s:  ______________                                            Number of 11s:  _______________

Number of 12s:  ______________                                            Number of 12s:  _______________

TOTAL SHOULD BE 25.                                                              TOTAL SHOULD BE 25.

Number of even #s:                                                                    Number of even #s:

 Roll # Total Parity Roll # Total Parity 51 76 52 77 53 78 54 79 55 80 56 81 57 82 58 83 59 84 60 85 61 86 62 87 63 88 64 89 65 90 66 91 67 92 68 93 69 94 70 95 71 96 72 97 73 98 74 99 75 100

For rolls #51 – 75:                                                                      For rolls #76 – 100:

Number of 2s:  ______________                                              Number of 2s:  _______________

Number of 3s:  ______________                                              Number of 3s:  _______________

Number of 4s:  ______________                                              Number of 4s:  _______________

Number of 5s:  ______________                                              Number of 5s:  _______________

Number of 6s:  ______________                                              Number of 6s:  _______________

Number of 7s:  ______________                                              Number of 7s:  _______________

Number of 8s:  ______________                                              Number of 8s:  _______________

Number of 9s:  ______________                                              Number of 9s:  _______________

Number of 10s:  ______________                                            Number of 10s:  _______________

Number of 11s:  ______________                                            Number of 11s:  _______________

Number of 12s:  ______________                                            Number of 12s:  _______________

TOTAL SHOULD BE 25.                                                              TOTAL SHOULD BE 25.

Number of even #s:                                                                    Number of even #s:

Create a probability frequency distribution using the numbers data above.

 Outcomes (Sum) Probability Rolls #1-25 Probability Rolls #1-50 Probability Rolls #1-75 Probability Rolls #1-100 What should the probability (expected value) be? 2 3 4 5 6 7 8 9 10 11 12 Total 1 1 1 1 1

Part A: Once you have completed the table, answer the questions below.

1. Did you use two dice or the random website?

Part B: Refer back to the probability frequency distribution table above.

1. Explain what you observe as you look at your group of rolls (sums) and compare them to the number in the last column.  (Look at the first column #1-25, then #1-50, etc.)

1. Explain what “Law of Large Numbers” mean.  If you did not get my explanation, google it.

1. Explain how the “Law of Large Numbers” was applied to this experiment.

1. Explain how you think this experiment would change if I asked you to roll the dice 1000 times.

1. How many times, do you think, you would have to roll to get to the expected value and explain why.

Part C: Create a probability frequency distribution using the parity data above.

 Outcomes (Parity) Probability Rolls #1-25 Probability Rolls #1-50 Probability Rolls #1-75 Probability Rolls #1-100 What should the probability (expected value) be? Even Odd

1. What was your longest streak (how many in a row), even or odd numbers? How many was this?

Refer back to the table giving the parity probability frequency.

1. Explain what you observe as you look at your group of rolls (parity) and compare them to the expected values.

1. When rolling 2 dice the sum of “7” is considered lucky.  Is there a lucky “even” or “odd” number?  Explain your answer.

Part D: Draw a bar graph using the sum of the dice for 100 rolls and the expected value (the last column) of the sum for rolling two dice. You can use the graph on the next page.

Draw another graph using the parity values. You can use the graph on the next page.

EXAMPLE:

1. Explain what you notice about the shape AND variation of the sum graph (Chapter 6).

1. Explain what you notice about the shape AND variation of the parity graph (Chapter 6).

Dice Rolls

Parity

RUBRIC FOR PROJECT:

Correct counting or rolls for each group of 25:  10%

Probability Frequency Tables:  50%

Questions (including grammar and punctuation):  40%

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