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Table of Contents
Abstract ............................................................................................... 3
Hypothesis .......................................................................................... 3
Materials ............................................................................................ 3
Procedure ............................................................................................ 3
Purpose ............................................................................................... 4
Research ............................................................................................. 4
What is Probability? ............................................................... 4
Where is Probability used? .................................................... 5
How long it took to do the Experiment ............................. 5
Possibilities with 2 dice ........................................................... 5
Predict Probability? .................................................................6
How do you test out Probability? .......................................6
How many sums? ...................................................................... 7
Recording the Results ............................................................. 7
Observations & Results ................................................................. 8
Conclusion......................................................................................... 10
Acknowledgments ........................................................................... 10
Bibliography .................................................................................... 10
Appendix (Data)............................................................................ 11
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Abstract
Why I chose this project is I wanted to know what are the most common sums of 2 dice. How I
did this was I rolled 2 dice and found out the most common sums. It was cool how I wasn’t totally
wrong. But I was still wrong. Why I picked this kind of a project is because math is fun and it still
has many more wonders that I don’t know about yet.
Hypothesis
I think 4s are going to come up the most often as the sum. I think the high numbers like 10, 11,
and 12 will come up the least.
Materials
1) Data sheets
2) 2 dice
3) A table
4) 2 pencils
Procedure
1) I rolled the dice
2) I recorded the numbers that came up on the dice
3) I added up the sum of the digits
4) I repeated that 100 times
5) I graphed the results
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Purpose
To figure out what are the most common sums of 2 dice.
Research
What is probability?
Probability is a kind of math that deals with the laws of chance. It is used to predict the most
likely outcome of uncertain events. Such as A is more probable than B. Or that getting a king of
spades in a normal deck of cards is less probable than getting a ace or a queen of spades which means
you will have a better chance of getting a queen or an ace than of getting a king (meaning if I wanted
a queen or an ace, and either one would do, I would have more of a chance of getting one or the other
more than a king). Or say I needed a 3, 4, 5, or a 6 to win my dice rolling game, but if I got a 1 or a
2 I would lose. What’s the probability of getting a 3,4, 5, or a 6? It’s 4/6, because out of 6 possibilities,
4 of them are winners, and it's only 2/6 of getting a 1 or a 2 because only 2 of the 6 possibilities are
losers.
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Where is Probability used?
It is used in insurance and also in physics. Insurance companies have to use it to calculate how
much they charge their customers. How they do it is if they calculate that a car has a 10% chance of
getting wrecked then they charge 10 people 1/10 of the value of the car so they can pay for the one car
that gets wrecked. It’s also used in predicting the weather. They use probability to tell us what would
probably happen tomorrow and next week. Such as what is the chance of snow on Tuesday?
How long it took to do the Experiment
I think you should do the experiment twice if you’re rolling dice like me. Roll the dice about 100
to 200 times. Another thing that is a variable is the number of dice that you roll. I rolled 2 dice and
I think that is the perfect number of dice to roll if you’re doing this experiment. For me it took about
30 minutes. I only rolled dice 100 times but if you roll the dice 200 times it would probably be about
1 hour. If you do it twice compare the results, see how close they are, and if you totally got weird
results you might have done something wrong.
Possibilities with 2 dice
There are 36 possibilities with 2 dice. They are: 1 and 1, 1 and 2, 1 and 3, 1 and 4, 1 and 5, 1 and
6, 2 and 1, 2 and 2, 2 and 3, 2 and 4, 2 and 5, 2 and 6, 3 and 1, 3 and 2, 3 and 3, 3 and 4, 3 and 5,
3 and 6, 4 and 1, 4 and 2, 4 and 3, 4 and 4, 4 and 5, 4 and 6, 5 and 1, 5 and 2, 5 and 3, 5 and 4,
5 and 5, 5 and 6, 6 and 1, 6 and 2, 6 and 3, 6 and 4, 6 and 5, and 6 and 6. And there are fewer sums
than possibilities. The sums are, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. There are only 11 sums.
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Predict Probability?
You can calculate probability and probability can be used to predict events. How probability
predicts is it gives you a general view of what might happen. Like if I wanted to know what the
weather is going to be like tomorrow, I could watch the Weather Channel to see what might happen,
or I could look at the newspaper in the morning. Each uses probability to get what it might be
tomorrow. A way they do this is they use old records to find out what it was like this time of year last
year. Also they send up little weather balloons that go really high in the sky and report on things like
temperature, humidity, and wind speed. In fact if I am correct 200 weather balloons go up every
hour. Satellites also send down reports that are used to calculate the probability of the kind of weather.
How do you test out probability?
A couple simple ways of testing probability are flipping a coin, rolling dice, or picking from a
hand of cards. I’ll show you how to do the dice experiment. How you would do it is get a die and roll
it 100 to 200 times. What’s the probability of getting a 6 on the die any time you roll it? It’s 1/6 just
like for 1, 2, 3, 4, and 5, because there are six sides on the die and each one is equally likely to come up.
(Dice, Coin,
And Cards picture)
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How many Sums?
How many sums of 5 are possible with 2 dice? 1 plus 4, 2+3, 4+1, and 3+2 are the four ways to
get five. Although 4+1 and 1+4 look the same, as well as 2+3 and 3+2, they are still 2 different
possibilities.
P
The four ways you can get a sum of 5.
Recording the Results
First, on a chart, I recorded what number came up on each die for each of the 100 trials. Then
I added up the two numbers to get the sum for each trial.
Tally marks are a good way of recording your results. How I did it is I made a chart like 2 then
down a line 3 down a line 4 and so forth on all the way to 12. Then I took tally marks of how many
times that sum came up and if you make it into a graph it shows it nicely. Graphs are fun and easy
to make.
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Observations and Results
Of the possible combinations there are 36 possibilities as shown in the chart below. However
there are only 11 sums, because there is more than one way to get some of the sums. So the probability
of getting a 2 is 1/36 because as shown in the chart below there’s only 1 combination that adds up to
2. But the probability of getting a 3 is 2/36 because there are 2 different combinatons that add up to
3. In this way, by counting how many times a sum is shown on the chart, and dividing by 36, you
can find the probability of getting that sum. So the probability of getting a 2 is 1/36 (2.8%), 3=2/36
(5.6%), 4=3/36 (8.3%), 5=4/36 (11.1%), 6=5/36 (13.8%), 7=6/36 (16.7%), 8=5/36 (13.8%), 9=4/36
(11.1%), 10=3/36 (8.3%), 11=2/36 (5.6%), and 12=1/36 (2.8%). The sum of all the probabilities is 1 or
100%. It is 100% because you have to get something on the chart!
Die #2
Die #1
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
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I rolled the dice 100 times. I got the following data for the different sums, which is shown on the
graph below. I got a sum of two 2 times (2%), a sum of three 7 times (7%), a sum of four 12 times
(12%), a sum of five 8 times (8%), a sum of six 21 times (21%), a sum of seven 17 times (17%), a sum
of eight 9 times (9%), a sum of nine 6 times (6%), a sum of ten 11 times (11%), a sum of eleven 5 times
(5%), and a sum of twelve 4 times (4%).
25
25
20
20
15
15
10
10
5
probability
in %
5
0
0
2
3
4
sum
5
6
the
or
7
8
9
10
eti
ca
l
ac
11
tua
12
ld
ata
My graph shows that I got a graph that was approximately the same shape as the graph of the
theoretical probabilities. Both graphs were short on both ends and tall in the middle. But my graph
was not perfect. I had more 4s and 10s than predicted, and fewer 5s, 8s, and 9s than predicted.
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I think that’s because I only rolled the dice 100 times. Probability tells you what probably will
happen, not what will happen. If I had rolled the dice 1000 times, the graphs would probably have
looked more alike.
Conclusion
I concluded that my hypothesis was wrong. Fours did not come up the most, 6s and 7s came up
the most. But I was right on the part that 10s, 11s, and 12s would come up the least. Also 2s and 3s
came up not very many times.
Acknowledgments
Thank you to my mom for suggesting this experiment, to my dad for helping me with my
internet research, to my sister Allison for lending me her dice, and to my mom for showing me how
and helping me to use Photoshop, ColorIt, Typestyler, DeltaGraph, and Pagemaker so that I could
do my report.
Bibliography
Adler, Irving, 1960. The Giant Golden Book of Mathematics: Exploring the World of Numbers
and Space. Golden Press, Western Publishing Co., New York.
Anonymous, 1997. Probability, in: Science Encyclopedia, 24 vols., JX: n. p.
Anonymous, date unknown. Probability. http://oldweb.uwp.edu/academic/mathematic/
probability/index.htm
Lanius, Cynthia, 2004. Let’s Do Math. http://www.math.rice.edu/~lanius/domath/dice.html
Linn, Charles F. , 1972. Probability. Thomas Y. Crowell Co., New York.
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Appendix (Data)
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