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Dice Rolls

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Dice Rolls

DICE ROLLS




Students in this course often ask why we study so much about gambling. Don't I know that gambling is an abomination unto the Lord!? My answer is yes! but so are the corporations and govenments that profit from our ignorance of gambling. But what is gambling? Gambling is simply playing with chance. Not necessarily a wise thing to do unless you are a casino or an insurance company. However, gambling games reveal the laws of chance in their simplest form. By studying dice games in Paris over 350 years ago, Pascal and Fermat invented the science of chance. Chance is our anglo-saxon word for probability. The mathematics of probability quickly begat the science of statistics. Today, statistics is the way we predict the future. Four hundred years ago, astrology, tarot cards, and crystal balls predicted the future, but not very well. Nowadays, we use numbers.

TWO DICE & TWO TABLES

 The table below shows all the ways you can add the spots on two 6-sided dice.
Columns A and B list all the ways the red and green dice can pair up.
Each row adds columns A and B, and puts their sum in the SUM column.


    
        

            
            
            D  
            I  
            E  
            
            A
            
        

        

            
            
            1
            2
            3
            4
            5
            6
        

        

            D
            1   
            2
            3
            4
            5
            6
            7
        

        

            I
            2    
            3
            4
            5
            6
            7
            8
        

        

            E
            3
            4
            5
            6
            7
            8
            9
        

        

            
            4
            5
            6
            7
            8
            9
            10
        

        

            B
            5
            6
            7
            8
            9
            10
            11
        

        

            
            6
            7
            8
            9
            10
            11
            12

The table below is a FREQUENCY TABLE.
This table shows the frequency and probability for each sum in the table above. The FREQUENCY column counts how many ways there are to roll each sum.
For example, there are 6 ways to roll the sum of 7.
The P(SUM) columns divide the frequency by 36, the total number of ways to roll 2 dice. The probability columns labelled P(SUM) show the probabilities as fractions and decimals. The sum of all the probabilities is, of course, 1, if you don't count the round off error.

 


    
        

            SUM
            FREQUENCY=ways to roll sum
            P(SUM)=frequency/totalways
            P(SUM)=decimal 
        

        

             1
             0
             0/36
             0
        

        

             2
             1
             1/36
             .028
        

        

             3 
             2
             2/36
             .056
        

        

             4
             3
             3/36
             .083
        

        

             5
             4
             4/36
             .111
        

        

             6
             5
             5/36
             .139
        

        

             7
             6
             6/36
             .167
        

        

             8
             5
             5/36
             .139
        

        

             9
             4
             4/36
             .111
        

        

             10
             3
             3/36
             .083
        

        

             11
             2
             2/36
             .056
        

        

             12
             1
             1/36
             .028
        

        

            SUMS
             
             36/36
            1.001




TWO MORE DICE — TWO MORE TABLES

Red Die Green Die



Click & Roll Dice


Assume now that you have two tetrahedral dice, each a 4-sided pyramid.
When rolling tetrahedral dice, we count the numbers the dice land on face-down.
Make a similar table as above for the sums of both dice.
Fill in the table below.



Then fill in the frequency table below for the probabilities of the sums.



QUESTIONS

  1. How many ways can you roll two 4-sided dice?
  2. What sum will you roll most often?
  3. How many ways can the most frequent sum be rolled?
  4. What is the probability of rolling this sum?
  5. What is the probability of NOT rolling this sum?
  6. What is the sum of the probability column titled P(SUM)?
  7. If you roll an 8 nine times in a row,
    what is the probablity you will roll an 8 on your tenth roll?
  8. What is the probability of rolling the sums 5 OR 7?
  9. What is the probability of rolling the sums 5 AND then 7?
  10. How many ways can you roll three 4-sided dice?
  11. If you roll four 4-sided dice,
    what is the probability AT LEAST one die will be a 4?
  12. If you roll two tetrahedral dice 24 times,
    what is the probability AT LEAST one roll will sum up to 8?

CHECK LIST

  1. Screencopy your two tables after completing them.
  2. Be sure to answer the questions above if you want any points.
  3. Show your math calculations.
  4. Email early and often to your instructor.
  5. Due time is noon on Thursday one week from assignment date.


Comments:

From wHolt - 10/21/08 8:26 AM

D  I  E  A
1 2 3 4 5 6
D 1   2 3 4 5 6 7
I 2    3 4 5 6 7 8
E 3 4 5 6 7 8 9
4 5 6 7 8 9 10
B 5 6 7 8 9 10 11
6 7 8 9 10 11 12

 

From wHolt - 10/20/08 1:57 PM

SUM FREQUENCY=ways to roll sum P(SUM)=frequency/totalways P(SUM)=decimal 
 1  0  0/36  0
 2  1  1/36  .028
 3   2  2/36  .056
 4  3  3/36  .083
 5  4  4/36  .111
 6  5  5/36  .139
 7  6  6/36  .167
 8  5  5/36  .139
 9  4  4/36  .111
 10  3  3/36  .083
 11  2  2/36  .056
 12  1  1/36  .028
SUMS       SUM OF PROBABILITIES =>  1.00 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

From wHolt - 10/20/08 1:42 PM

 


	
		

			Die A
			1
			1 
			1 
			1 
			1 
			1 
			2 
			2 
			2 
			2 
			2 
			2 
			3 
			3 
			3 
			3 
			3 
			3 
			4 
			4 
			4 
			4 
			4 
			4 
			5 
			5 
			5 
			5 
			5 
			5 
			6 
			6 
			6 
			6 
			6 
			6 
		

		

			Die B
			1 
			2 
			3 
			4 
			5 
			6 
			1 
			2 
			3 
			4 
			5 
			6 
			1 
			2 
			3 
			4 
			5 
			6 
			1 
			2 
			3 
			4 
			5 
			6 
			1 
			2 
			3 
			4 
			5 
			6 
			1 
			2 
			3 
			4 
			5 
			6 
		

		

			Sum
			2
			3 
			4 
			5 
			6 
			7 
			3 
			4 
			5 
			6 
			7 
			8 
			4 
			5 
			6 
			7 
			8 
			9 
			5 
			6 
			7 
			8 
			9 
			10
			6 
			7 
			8 
			9 
			10
			11
			7 
			8 
			9 
			10
			11
			12

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Last Modified 10/26/08 8:31 AM