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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 governments 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 gambling 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 math.

ASSIGNMENT

Create a table like the one below. 
Insert as many rows as there are ways to roll two 6-sided dice.
In columns A and B, list all the ways the spots can land face up when you roll two 6-sided dice.
How many ways are there? And which counting rule tells you?
On each row, add columns A and B, and put their sum in the SUM column.
Count the number of ways you can roll each SUM.
You do not need to show us this table, but you will need its data for the next table.

Die A Die B SUM
 1 1 2
 1 2 3
   
   
After the table above is completed, copy the table below into a comment, and fill it in.
We call this type of table a Frequency Table.

Using the results from the table above, count how many ways there are to roll each sum.
Enter this number in the FREQUENCY column next to the corresponding SUM.
For example, if there are 10 ways to roll the sum of 7, enter 10 on row 7.









In the P(SUM) column, divide the frequency by the total number of ways to roll 2 dice.
Add all the probabilities in the P(SUM) column, and enter the number in the bottom row.
Can you guess what the sum of the probability columns will be?
SUMFREQUENCY=ways to roll sumP(SUM)=frequency/totalwaysP(SUM)=decimal 
 1   
 2   
 3    
 4   
 5   
 6   
 7   
 8   
 9   
 10   
 11   
 12   
  SUM OF PROBABILITIES => ?  

 

 

 

 

 

 

 

 

 

 

 

 

  1. Copy the blank table above to a comment, and complete it.
  2. Submit your comment.
  3. Hoversnap your comment.
  4. Delete your comment.
  5. Send me your Hoversnap table privately as a gif attachment.

    QUESTIONS

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?    
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		  
3. What is the probability of rolling a 7?    
4. What is the probability of NOT rolling a 7?   
5. What is the sum of the last two probabilities?   
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		  

7. What is the probability of rolling 7 OR 11?  

  

8. What is the probability of rolling 7 AND then 11? 

  
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		  

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

Research these questions on the web
Show us HOW you got your answers using + – x ÷
Email me the answer to #10 privately.
You receive no points till you answer #10.
Give answers in both fractions and decimals.
The frequency table is worth 4 points and the questions 10.
That's 14 points altogether. WOW!

Comments:

From wHolt - 12/15/06 5:22 PM

TBird - you have about 4 different versions of these solutions.
Which one do you want to go with?
I will score the last version when you answer #7.

From TBird - 12/15/06 2:19 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?  366 Numbers on first die*6 Numbers on second
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		61+6,2+5,3+4,4+3,5+2,6+1
3. What is the probability of rolling a 7?  .167=16.7%6 Ways to roll a seven / 36 total ways.
4. What is the probability of NOT rolling a 7? .834=83.4%1-.167, or (1/36)x2+(2/36)x2+(3/36)x2+(4/36)x2+(5/36)x2
5. What is the sum of the last two probabilities? 1.001.834+.167
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		.167=16.7Probability does not change based on the number of rolls.

7. What is the probability of rolling 7 OR 11?  

.222=22.2%6/36 prob. of rolling a seven + 2/36 prob. of rolling eleven=8/36

8. What is the probability of rolling 7 AND then 11? 

.00926=.926%(6/36)x(2/36)=.00926
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		.069=6.9%

Off the wall here: 1 roll out of 24 is a 12 which is 1/36=

1/24+1/36=.0694

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From Spartan - 12/14/06 11:08 AM

Questions 

Answer 

How? 

1. How many ways can you roll two 6-sided dice?  36 2 dice  6 outcomes each (6x6)

2. How many ways can you roll a 7?

 6 1,6  2,5  3,4  4,3  5,2  6,1
3. What is the probability of rolling a 7?  1/6 6/36 = 1/6
4. What is the probability of NOT rolling a 7?  5/6 30/36 = 5/6
5. What is the sum of the last two possibilities?  1 1/6 + 5/6 = 1

6. If you roll a 7 nine times in a row, what is your probability

you will roll a 7 your tenth time? 

 1/6 Each roll has the same probablility as the one before it.
7. What is the probability of rolling a 7 or an 11?  2/9 1/6 + 1/18 (3/18 + 1/18 = 4/18 = 2/9)
8. What is the probability of rolling a 7 and then an 11?  1/108 1/6 x 1/18 = 1/108

9. If you roll two dice 24 times, what is the probability AT LEAST 

one roll will be a 12? 

 2/3 1/36 24 times (1/36 x 24/1 = 2/3)

10. If you roll 4 dice, what is the probability AT LEAST one die will

be a 6?

  emailed

From Houdini - 12/11/06 8:05 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?  366 Numbers on first die*6 Numbers on second
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		61+6,2+5,3+4,4+3,5+2,6+1
3. What is the probability of rolling a 7?  .167=16.7%6 Ways to roll a seven / 36 total ways.
4. What is the probability of NOT rolling a 7? .834=83.4%1-.167, or (1/36)x2+(2/36)x2+(3/36)x2+(4/36)x2+(5/36)x2
5. What is the sum of the last two probabilities? 1.001.834+.167
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		.167=16.7Probability does not change based on the number of rolls.

7. What is the probability of rolling 7 OR 11?  

.222=22.2%6/36 prob. of rolling a seven + 2/36 prob. of rolling eleven=8/36

8. What is the probability of rolling 7 AND then 11? 

.00926=.926%(6/36)x(2/36)=.00926
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		.069=6.9%

Off the wall here: 1 roll out of 24 is a 12 which is 1/36=

1/24+1/36=.0694

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From Cheana - 12/5/06 9:59 PM

QUESTIONS
QUESTIONS ANSWER HOW?
1. How many ways can you roll two 6-sided dice? 36 ways 6*6      
2. How many ways can you roll a 7? 6 ways
(How many ways can the spots add up to 7?)       
3. What is the probability of rolling a 7?  0.16666   6/36= 
4. What is the probability of NOT rolling a 7?      0.833287        1-0.16666=
5. What is the sum of the last two probabilities?  1 (or .999887)   
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  6/36=0.1666     
7. What is the probability of rolling 7 OR 11?  0.1666 + 0.0555= 0.22215
    
8. What is the probability of rolling 7 AND then 11?  0.1666* 0.0555= 0.7221
    
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?      1-.0277 (the probability of rolling a 12)= .9723^24= .5095735273 then 1- .5095735273= 0.4913062167

From DirtyBird - 12/5/06 7:50 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 6*6=36
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6                         [1,6] [2,5] [3,4] [4,3] [5,2] [6,1] 
3. What is the probability of rolling a 7?   1/6 or 16.66% 6/36=.1666
4. What is the probability of NOT rolling a 7?  5/6 or 83.33% 30/36=.8333
5. What is the sum of the last two probabilities?  1 1/6+5/6=6/6=1
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 1/6 or 16.66% 6/36=.1666

7. What is the probability of rolling 7 OR 11?  

 22.21%

 Rollin 7's=.1666

Rollin 11's=.0555

.1666+.0555=.2221

8. What is the probability of rolling 7 AND then 11? 

 .93% .1666*.0555=.0924
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 49% 1-(35/36)^24=1-.97^24=.4914

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

 

From Fro - 12/5/06 7:00 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36

 6 * 6 =36

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6

 [1&6] [6&1] [2&5] [5&2] [3&4] [4&3]

3. What is the probability of rolling a 7?  .1666 or 16% 1/6 = .1666 or 16%
4. What is the probability of NOT rolling a 7?  .8333 or 83% 5/6 + 5/6 + 5/6 + 5/6 + 5/6 + 5/6 = 30/36 = .8333
5. What is the sum of the last two probabilities?  1 or 100%

 1/6 + 5/6 = 6/6 = 1

6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		.1666 or 16%

 6/36. Previous rolls do not change the probability of the next roll. 

7. What is the probability of rolling 7 OR 11?  

 .222 or 22%

 6/36 = (of rolling a 7)  2/36 (of rolling an 11) = 8/36 = .2222

8. What is the probability of rolling 7 AND then 11? 

 1/108 or .00925

 1/6 (chance of rolling 7)
1/18 (chance of rolling 11)
1/6 * 1/18 (to find chance of rolling 7 AND 11) = 1/108 

9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		.0277 or 2.7%

 1/36 probability of rolling 12. The number of rolls does not change the probability of rolling a 12. There is only 1 way to roll a 12 with 2 die.

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From David - 12/4/06 5:52 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   366 sides/die x 2 dice or 6*6=36 
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 1,6-2,5-3,4-4,3-5,2-6,1
3. What is the probability of rolling a 7?   .1666 or 16.7% or 1:6  (form question two)/36 (from question one)=.16666
4. What is the probability of NOT rolling a 7?  .8333 or 83.3% or 5:6 or 1:1.2(so many ways!) 30/36
5. What is the sum of the last two probabilities?  1 .166+.833=1
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 .1666 or 16.7% or 1:6 nothing changes, you have as good of chance rolling a 7 on your first, tenth, or hundreth roll.

7. What is the probability of rolling 7 OR 11?  

  .222 or 22.2% or 1:4.5 or 2:9 Take the chances of rolling a seven from above and add to that the chances of rolling an eleven.  Possible ways to roll an eleven 6,5-5,6.

8. What is the probability of rolling 7 AND then 11? 

 .00925 or .92% or 1:108 AND indicates that you must multiply the two possibilities together.
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 .4914 or 49.14% or 1:2.05 like some of the students before me, it is easier to find the possibilities of what will NOT happen.

From TBird - 12/4/06 1:55 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36

 6 dice * 6 outcomes of each

=36

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6

 {1,6},{2,5},{3,4},

{6,1},{5,2},{4,3}

3. What is the probability of rolling a 7?   1/6 or .1666 6/36 = 1/6 = .1666
4. What is the probability of NOT rolling a 7?  5/6 or .8333 30/36 = 5/6 = .8333
5. What is the sum of the last two probabilities?  1

 1/6 + 5/6

6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 1/6 or .1666

 6/36. Six ways to roll a seven out of 36 ways to roll two 6 sided dice. The rolls
previously made do not change the probabilit of the next roll. 

7. What is the probability of rolling 7 OR 11?  

 2/9 or .222

 6/36 = 1/6 (of rolling a 7)+ 2/36 = 1/18 (of rolling an 11) = 4/18 = 2/9.

8. What is the probability of rolling 7 AND then 11? 

 1/108 or .00925

 1/6 (chance of rolling 7)
1/18 (chance of rolling 11)
1/6 * 1/18 (to find chance of rolling 7 AND 11) = 1/108 

9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		.4914

 1/36 probability of rolling 12
1-(35/36)^24 = .4914 

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From SuperDuke - 12/2/06 10:35 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 rolls 6*6=36 but of that there are only 21 different combinations 6+5+4+3+2+1=21 it is not 36 because 15 of the rolls will be the same numbers.  
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 ways 1+6, 2+5, 3+4, 4+3, 5+2, 6+1
3. What is the probability of rolling a 7?   1/6 or 16.66% 6 combinations (of 7) out of 36 possible rolls. 6:36 = 1:6 = 1/6 = 16.66%
4. What is the probability of NOT rolling a 7?  5/6 or 83.33%24 combinations (of not 7) out of 36 rolls. 24:36 = 5:6 = 5/6 = 83.33%
5. What is the sum of the last two probabilities?  100% the sum of probabilities must equal 1
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 1/6 or 16.66%on the tenth roll 6 out of the thirty six possible rolls will add up to 7. so it is 6 out of 36 = 6/36 = 1/6 = .1666 = 16.66%

7. What is the probability of rolling 7 OR 11?  

 22.21% you have a 5.55% chance of crappin out on a 11 (2/36) + a 16.66% chance of catchin a 7.

8. What is the probability of rolling 7 AND then 11? 

 .92%  5.55% chance of rolling an 11 after a 16.66% chance of rollin a 7.  16.66*.0555 = .92%
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 49%After much heartache I have determined this is correct. The reason it is correct is because of the infinitey of the number. You cannot reach 1 because you are always "rolling" an odd that is a fraction of 1 and the product of those fractions is your odds so to speak.. This is the reason u can never garantee any number or coin flip.  So here it is 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 *35/36 *35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36=.5086 This is ruffly a 51% chance to NOT hit. so we have a 49% chance TO hit

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

 

From wHolt - 12/2/06 1:03 PM

I have written this a few times now about #9:

Does this mean that if rolling six times, it is certain you will roll a 6?
1/6*6=1
would you bet your life on it?
how about rolling 7 times?
what does 7/6 of a chance mean?
greater than certain?
the problem with this method is that you get nonsensical results.
try another approach.
read other comments.

From Pac - 12/2/06 10:53 AM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 6 ways, 2 dice = 6^2=36
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 (1,6);(2,5);(3,4);(4,3);(5,2);(6,1)
3. What is the probability of rolling a 7?   0.166667 or 16.67% 6 ways to get a 7, 36 total possibilities = 6/36=0.166667
4. What is the probability of NOT rolling a 7?  0.833333 or 83.33% 30 ways NOT to get a 7, 36 total ways = 30/36=0.833333
5. What is the sum of the last two probabilities?  1 or 100% 0.166667+0.833333=1 or 16.67%+83.33%=100%
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 0.166667 or 16.67% It doesn't matter what you rolled the previous nine times because the probability of rolling a 7 with two dice is always going to be 16.67%.

7. What is the probability of rolling 7 OR 11?  

 0.222222 or 22.22% probability of 7 = 0.166667; probability of 11 = 0.055556; 0.166667+0.055556=0.222222 or 22.22%

8. What is the probability of rolling 7 AND then 11? 

 0.009259 or 0.9259% probability of 7 = 0.166667; probability of 11 = 0.055556; 0.166667*0.055556=0.009259 or 0.9259%
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 0.666667 or 66.67% probability of 12 = 0.027778; 24 dice rolls = 0.027778*24=0.666667 or 66.67%

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From Sunshine - 11/30/06 12:43 AM

1. How many ways can you roll two 6-sided dice?  36 6 x 6
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 .1667 x 36
3. What is the probability of rolling a 7?   16.67% 6 /36
4. What is the probability of NOT rolling a 7?  83.33% 30/36
5. What is the sum of the last two probabilities?  1 .8333 + .1667
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 16.67% 6/36

7. What is the probability of rolling 7 OR 11?  

 22.23% .1667 = .0056

8. What is the probability of rolling 7 AND then 11? 

 .93%.1667 x .0556 
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 27.8% 24/36

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From Draco - 11/29/06 8:23 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 6*6
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 .1667*36
3. What is the probability of rolling a 7?   .1667 or 16.67% 6/36
4. What is the probability of NOT rolling a 7?  .8333 or 83.33% 1-.1667
5. What is the sum of the last two probabilities?  1 .8333+.1667
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 .1667 or 16.67% 6/36

7. What is the probability of rolling 7 OR 11?  

 .2223 or 22.23% .1667+.0556

8. What is the probability of rolling 7 AND then 11? 

 .0093 or .93% .1667*.0556
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 .278 or 27.8% This is the probability of rolling a 12 AT LEAST once: 1/36

From Melewen - 11/29/06 7:45 PM

QUESTIONS
ANSWER
HOW?
1. How many ways can you roll two 6-sided dice?
36 6 possibilities on the first x 6 possibilities on the second = 36
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)
6 1,6 ; 6,1 ; 2,5 ; 5,2 ; 3,4 ; 4,3 
3. What is the probability of rolling a 7?
17% 6 ways to roll a 7 divided by the total possibilities of 36; 6/36 = .16777777
4. What is the probability of NOT rolling a 7?
83% 100% - 17% = 83%. or 1-.167777 = .83333
5. What is the sum of the last two probabilities?
100 

Of course they have to add up to 100 (or 1), because you either roll a
seven or you don't. Tautology! 

6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?
17% 

Your tenth roll has nothing to do with the past nine (unless they're
weighted or something), so the probability is the same. Gambler's fallacy!

7. What is the probability of rolling 7 OR 11? 
22%
There are two ways to roll an eleven. Add those two ways to the six ways to
roll a seven, that makes eight, and you divide it by the total possibilities, 36.
8/36 = .2222, or 22%
8. What is the probability of rolling 7 AND then 11?
0.9258% Take the probability of rolling a seven, .1666, and multiply it by the probability
of rolling an eleven, .055555, which gives you .009258.
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?
66% The probability of rolling a 12, times the number of times you're rolling, 24,
which equals .6666666, or 66%
10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?
PRIVATE EMAIL

From Kathi - 11/29/06 9:32 AM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36

 6 dice * 6 outcomes of each

=36

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6

 {1,6},{2,5},{3,4},

{6,1},{5,2},{4,3}

3. What is the probability of rolling a 7?   1/6 or .1666 6/36 = 1/6 = .1666
4. What is the probability of NOT rolling a 7?  5/6 or .8333 30/36 = 5/6 = .8333
5. What is the sum of the last two probabilities?  1

 1/6 + 5/6

6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 1/6 or .1666

 6/36. Six ways to roll a seven out of 36 ways to roll two 6 sided dice. The rolls
previously made do not change the probabilit of the next roll. 

7. What is the probability of rolling 7 OR 11?  

 2/9 or .222

 6/36 = 1/6 (of rolling a 7)+ 2/36 = 1/18 (of rolling an 11) = 4/18 = 2/9.

8. What is the probability of rolling 7 AND then 11? 

 1/108 or .00925

 1/6 (chance of rolling 7)
1/18 (chance of rolling 11)
1/6 * 1/18 (to find chance of rolling 7 AND 11) = 1/108 

9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		.4914

 1/36 probability of rolling 12
1-(35/36)^24 = .4914 

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From Zonino - 11/28/06 10:57 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 There are 6 ways you can roll the first dice.  There are 6 ways you can roll the second dice.  6*6 = 36.
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 You can roll the following: 1 and 6, 6 and 1, 3 and 4, 4 and 3, 5 and 2, 2 and 5.
3. What is the probability of rolling a 7?   0.16667 or 16.667% There are 6 ways of rolling a 7 out of 36 possible ways to roll the dice total.  6/36 is 0.16667 or 16.667%
4. What is the probability of NOT rolling a 7?  0.83333 or 83.333% If there are 6 ways to roll a 7 then there are 30 ways to not roll a 7.  30/36 = 0.83333 or 83.333%.  The other way is to subtract the probability of rolling a 7 from 1.
5. What is the sum of the last two probabilities?  1 or 100%. 0.83333 + 0.16667
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 0.16667 or 16.667% Each die roll does not depend on the previous rolls.  Thus there are still 6 ways to roll a 7 out of 36 possible ways.  6/36 is 0.16667 or 16.667%

7. What is the probability of rolling 7 OR 11?  

 0.22222 or 22.222% There are 6 ways to roll a 7 and 2 ways to roll a 11.  So there are 8 possible ways total to roll a 7 or 11.  8 ways out of 36 total possibilities is 8/36 or 0.22222 or 22.222%

8. What is the probability of rolling 7 AND then 11? 

 0.009259 or 0.9259%. There are 6 ways to roll a 7 and 2 ways to roll a 11.  6/36 ways * 2/36 ways equals 12/1296 or 0.009259 or 0.9259%.
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 0.4914 or 49.14% You can roll a 12 1 out of 36 ways.  You can not roll a 12 35 out of 36 ways.  If you roll 24 times it would be (35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 *35/36 *35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36 * 35/36)  This is .5086.  We subtract from 1 to find the probability of rolling at least a 12.  1-.5086 = 0.4914 or 49.14%

From Pringle - 11/28/06 10:42 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 (1,1)(1,2)(1,3)(1,4)(1,5)(1,6)(2,1)(2,2)(2,3).....(6,4)(6,5)(6,6) => 6*6=36
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 (1,6) (2,5) (3,4) (4,3) (5,2) (6,1)
3. What is the probability of rolling a 7?   6/36(=0.0167) there are 36 total ways to roll 2 dice. So I divide 6 by 36
4. What is the probability of NOT rolling a 7?  30/36(=0.8333) there are 30 ways not to roll a 7 out of 36 (36-6=30)
5. What is the sum of the last two probabilities?  36(1)

 6+30=36

 0.0167+0.8333=1

6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  

 6/36

(=0.0167)

  I can have 6 chances   to roll 7 out of 36 ways. It doesn't matter how many times I roll a 7. It doesn't reduce or increase my chance to roll a 7 again.

7. What is the probability of rolling 7 OR 11?  

 8/36

(=0.2222)

 6/36 (probability of roll a seven)+2/36(probablility of roll 11)= 8/36

8. What is the probability of rolling 7 AND then 11? 

 1/108

(=0.0093)

6/36 (=0.0167 probability of roll a 7)

* 2/36 (= 0.0556 probability of roll a 11) = 12/1296= 1/108 =0.0093

9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  

 

0.0278

 

You have 36 ways to roll two dice and you can roll dice 24 times   =>   36*24= 864 -> total ways

 And you have 1 way to roll a 12 out of 36 and 35 ways not to roll a 12 out of 36 ways and you roll dice 24 times 35*24=  840

 The probability of not rolling a 12 is 840/864=0.97222  and the probability of at least one roll is a 12 is 1(sum of all the probabilities)- 0.97222= 0.0278 or (864-840)/864

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

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From Tiger - 11/28/06 9:28 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 6*6  6 ways to roll each one
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 (1,6) (6,1) (2,5) (5,2) (3,4) (4,3)
3. What is the probability of rolling a 7?   16.6% 6/36=.166666
4. What is the probability of NOT rolling a 7?  83.3% 30/36=.833333
5. What is the sum of the last two probabilities?  1 .16666666+.833333333=1
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 1/6 It is the same every time.

7. What is the probability of rolling 7 OR 11?  

 8/36 2/36+6/36=8/36=22.2%

8. What is the probability of rolling 7 AND then 11? 

 .925% 2/36*6/36=12/1296=.009259
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 66.6% 1/36*24=.666666  1 way to roll 12 but you have 24 chances

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

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From Bubba - 11/28/06 8:13 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 6*6
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 1,6 2,5 3,4 4,3 5,2 6,1
3. What is the probability of rolling a 7?   .1666 6/36
4. What is the probability of NOT rolling a 7?  .8334 30/36
5. What is the sum of the last two probabilities?  1 .1666 + .8334
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 .0000000164 .1666 to the seventh power

7. What is the probability of rolling 7 OR 11?  

 .222 8/36

8. What is the probability of rolling 7 AND then 11? 

 .0092592 .1666*.0555
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 .666 24/36

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

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From 7Iron - 11/28/06 7:46 PM

QUESTIONS
ANSWER
HOW?
1. How many ways can you roll two 6-sided dice?  
 36
 6*6
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  
 6
 1,6;2,5; 3,4;4,3;5,2;6,1
3. What is the probability of rolling a 7?  
 .1667
 6/36
4. What is the probability of NOT rolling a 7?
 .8333
 1- (6/36)
5. What is the sum of the last two probabilities? 
 1
 (6/36) +  (1-(6/36))
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  
 6/36
 Die rolls are independent of each other therefore it does not matter what was rolled previously because it does not determine the next outcome.
7. What is the probability of rolling 7 OR 11?  
 8/36 = 2/9
 (6/36) + (2/36) = 8/36 = .2222
8. What is the probability of rolling 7 AND then 11? 
 1/108
 (6/36) *(2/36) = 1/108 = .0093
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  
 4.4539*10^-38
 (1/36) * (1/36) … 24 times or (1/36)^24

From Trixie - 11/28/06 7:11 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 6*6 = 36
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 2,5  3,4  6,1  5,2  4,3 1,6
3. What is the probability of rolling a 7?   16.67% 6/36 = .1666 = 16.67%
4. What is the probability of NOT rolling a 7?  83.33% 30/36 = .8333 = 83.33%
5. What is the sum of the last two probabilities?  100% 83.33% + 16.67% = 100%
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 16.67% 6/36 = .1666 = 16.67%

7. What is the probability of rolling 7 OR 11?  

 22.22%

 6/36 + 2/36 = 8/36 = .2222 = 22.22% (36 total possibilites) (6 possibilites of rolling a 7) (2 possibilities of rolling an 11)

8. What is the probability of rolling 7 AND then 11? 

 .93% 6/36 * 2/36 = .00925 = .93%
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 66.67% 24/36 = .6667 = 66.67%

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

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From BassLady - 11/28/06 1:39 PM

    QUESTIONS

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36

1,1 - 1,2 - 1,3 - 1,4 - 1,5 - 1,6 - 2,1 - 2,2 - 2,3 - 2,4 - 2,5 - 2,6 - 3,1 - 3,2 - 3,3 - 3,4 - 3,5 - 3,6 - 4,1 - 4,2 - 4,3 - 4,4 - 4,5 - 4,6 - 5,1 - 5,2 -5,3 - 5,4 - 5,5 - 5,6 - 6,1 - 6,2 - 6,3 - 6,4 - 6,5 - 6,6 

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 61 and 6, 6 and 1, 2 and 5, 5 and 2, 4 and 3, 3 and 4 
3. What is the probability of rolling a 7?   1 out of 636 different rolls and chances of 7 are 6 out of 36 and if you reduce the 6/36 fraction it is 1/6 
4. What is the probability of NOT rolling a 7?  5 out of 636 different rolls and 30 of them no chance of 7.  If you reduce the 30/36 fraction it is 5/6. 
5. What is the sum of the last two probabilities?  1 6/6 is 1
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 1 out of 6 No matter how many times you roll the dice, the next roll your odds are back to the same of being 1 out 6.  There are only 6 chances of 36 different rolls.

7. What is the probability of rolling 7 OR 11?  

 8 out of 36 or 4 out of 9 if you reduce There are 6 ways to roll a 7 and 2 ways to roll an 11.  That is 9 of 36 odds.

8. What is the probability of rolling 7 AND then 11? 

 1 out of 18 If you roll a 7 your chances of rolling an 11 next remain the same as before.  You still have a 2 out of 36 or 1 of 18 chance.
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 1.5Odds of rolling a 12 are 1 in 36. 36 / 24 is 1.5

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From GolfGirl - 11/28/06 12:10 PM

 

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 ways

6^2= 36

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 ways1,6  4,3  5,2  6,1  3,4  2,5

3. What is the probability of rolling a 7?   16.67% 6/36 = 0.166667 = 16.67%

4. What is the probability of NOT rolling a 7?  83.33% 30/36 = 0.8333333 = 83.33%

5. What is the sum of the last two probabilities?  100% 

16.67% + 83.33% = 100.00%
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 16.67%
 6/36 = 1/6 = 0.166667 = 16.67%

7. What is the probability of rolling 7 OR 11?  

 22.22% 6/36 = 1/6, 2/36 = 1/18   (1/6)+(1/18) = 0.222222 = 22.22%

8. What is the probability of rolling 7 AND then 11? 

 0.93%

 6/36 = 1/6, 2/36 =  (1/6)(1/18) = 0.00925 = 0.93%
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 66.67% (1/36)(24) = 0.666667 = 66.67%

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

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From Phoenix - 11/28/06 11:29 AM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 6*6=36
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6

 1+6,2+5,3+4

6+1,5+2,4+3

3. What is the probability of rolling a 7?   1/6 or .1666 6/36
4. What is the probability of NOT rolling a 7?  5/6 or .8333 30/36
5. What is the sum of the last two probabilities?  1 6/36 + 30/36
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 1/6 or .1666 6/36. Six ways to roll a seven out of 36 ways to roll two 6 sided dice.

7. What is the probability of rolling 7 OR 11?  

 2/9 or .222 6/36 + 2/36 = 8/36 or 2/9. Six ways to roll a seven. Two ways to roll an eleven.

8. What is the probability of rolling 7 AND then 11? 

  
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 .666 or 2/3 1 * 24 / 36. 1 way to roll a twelve 24 times out of 36 ways.

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

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From wHolt - 11/27/06 10:00 AM

Harkar - read my comment on 11/25

From Harkar - 11/26/06 8:25 PM

QUESTIONS
ANSWER
HOW?
1. How many ways can you roll two 6-sided dice?  
 36
 6*6: Two dice with 6 possible outcomes for each dice.
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  
 6 ways
 1 & 6, 2 & 5, 3 & 4, 4 & 3, 5 & 2, 6 & 1
3. What is the probability of rolling a 7?  
 .1666 = 16.7%
6/36 = .1666
4. What is the probability of NOT rolling a 7?
.8333 = 83.33%
 30/36 = .8333
5. What is the sum of the last two probabilities? 
 1 = 100%
.1666 + .8333 = .9999 (1)
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  
.1666 = 16.7%
The probability remains the same for each roll, regardless of past outcomes.
7. What is the probability of rolling 7 OR 11?  
 .222 = 22.2%
 8/36 = .222 = 22.2%
8. What is the probability of rolling 7 AND then 11? 
12/1296 = 0.93%
 6/36 * 2/36 = 12/1296 = .0093
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  
0.02778 = 2.778%
 1/36 * 24/24 = 0.02778 = 2.778%

From wHolt - 11/25/06 12:44 PM

Folks - if you roll the dice 36 times,
is it certain that you will roll a 7 at least once?
That's what some of your methods predict.
The "at least" should be a clue on how to think about #9.

From Kathi - 11/25/06 9:25 AM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36  

 Two dice 6 numbers of each:

6 * 6 = 36 

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6

It will take both dice to make a 7. In which case there is only 6 ways ( Notice that the order does matter with two different die):


1 & 6, 2 & 5, 3 & 4, 4 & 3, 5 & 2, 6 & 1
 

3. What is the probability of rolling a 7?   .1666 6/36 =  .1666
4. What is the probability of NOT rolling a 7?  .8334 .1666 * 5 =  .8334
5. What is the sum of the last two probabilities?  1 .1666 + .8334 = 1
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 .00000001647 There is .1666 probability of rolling a 7, so .1666 * .1666* .1666* .1666* .1666* .1666* .1666* .1666* .1666* .1666* = .00000001647

7. What is the probability of rolling 7 OR 11?  

 .2222

 There are 8 ways to roll either a 7 or 11 and 36 possible outcomes of the dice.

8/36 =  .2222

 

8. What is the probability of rolling 7 AND then 11? 

 .02775556 There is .1666 probability of rolling a 7 and there is a .1666 probability of rolling an 11; 1666* .1666= .02775556
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 .6667 Twentyfour rolls and 36 possible out comes:
24/36= .6667

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

This is a correction for #7 

From CenterField - 11/24/06 10:06 PM

QUESTIONS

ANSWERHOW?
1. How many ways can you roll two 6-sided dice?  

If you count order...
36

6*6: Two positions with 6 possible outcomes for each position.
2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		6

1+6, 2+5, 3+4, 4+3, 5+2, 6+1

3. What is the probability of rolling a 7?   16.7% 6/36
4. What is the probability of NOT rolling a 7?  83.3% 30/36
5. What is the sum of the last two probabilities?  1, or 100%The sum of the probabilty of something happening plus the probability of it not happening always equals 1, or 100%.
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 16.7%The probability remains the same for each roll, regardless of past outcomes.

7. What is the probability of rolling 7 OR 11?  

 22.2% 6/36 + 2/36 = 8/36 = .222

8. What is the probability of rolling 7 AND then 11? 

 0.92% 6/36 * 2/36 = 12/1296 = .009259
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 66.7% 1/36 * 24 = 24/36 = .666667

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

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From TBird - 11/24/06 5:00 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36 ways

6^2= 36

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6 ways1,6  2,5  3,4  4,3  5,2  6,1

3. What is the probability of rolling a 7?   16.67% 6/36 = 1/6 = 0.166667 = 16.67%

4. What is the probability of NOT rolling a 7?  83.33% 30/36 = 5/6 = 0.8333333 = 83.33%

5. What is the sum of the last two probabilities?  100% 

16.67% + 83.33% = 100.00%
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 16.67%
 6/36 = 1/6 = 0.166667 = 16.67%, The probability will always be the same as the first roll.


7. What is the probability of rolling 7 OR 11?  

 22.22% 6/36 = 1/6, 2/36 = 1/18   (1/6)+(1/18) = 0.222222 = 22.22%

8. What is the probability of rolling 7 AND then 11? 

 0.93%

 6/36 = 1/6, 2/36 = 1/18  (1/6)(1/18) = 0.00925 = 0.93%
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 66.67% (1/36)(24) = 0.666667 = 66.67%

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From wHolt - 11/24/06 4:53 PM

Boki -
on #6, the question does not ask you for the probability of rolling ten 7's in a row.
read it more carefully.

Kathi-
show us how you are figuring #7

From Kathi - 11/24/06 10:13 AM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   36  

 Two dice 6 numbers of each:

6 * 6 = 36 

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 6

It will take both dice to make a 7. In which case there is only 6 ways ( Notice that the order does matter with two different die):


1 & 6, 2 & 5, 3 & 4, 4 & 3, 5 & 2, 6 & 1
 

3. What is the probability of rolling a 7?   .1666 6/36 =  .1666
4. What is the probability of NOT rolling a 7?  .8334 .1666 * 5 =  .8334
5. What is the sum of the last two probabilities?  1 .1666 + .8334 = 1
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 .00000001647 There is .1666 probability of rolling a 7, so .1666 * .1666* .1666* .1666* .1666* .1666* .1666* .1666* .1666* .1666* = .00000001647

7. What is the probability of rolling 7 OR 11?  

 .1388 5/36 =  .1666

8. What is the probability of rolling 7 AND then 11? 

 .02775556 There is .1666 probability of rolling a 7 and there is a .1666 probability of rolling an 11; 1666* .1666= .02775556
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 .6667 Twentyfour rolls and 36 possible out comes:
24/36= .6667

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

PRIVATE EMAIL

From Boki - 11/23/06 11:02 AM

QUESTIONS
ANSWER
HOW?
1. How many ways can you roll two 6-sided dice?  
 36

The set of all possible outcomes is: {1,2,3,4,5,6} for 1 6-sided dice.

For two dice will be: 6*6=36.

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  
 6
(1,6) (6,1) (2,5) (5,2) (3,4) (4,3)
 3. What is the probability of rolling a 7?  
 1/6
  6/36= 1/6=
0.16666666666666666666666666666667
4. What is the probability of NOT rolling a 7?
 5/6
 5/6=
0.833333333333333333333333333334
5. What is the sum of the last two probabilities? 
 1
 1/6+5/6=6/6=1
6. If you roll a 7 nine times in a row,
what is the probability you will roll 7 on your tenth roll?  
1/6

 the probability that 7 will come up on a tenth is 1/6.

No matter how many times we roll a die, the probability that 7

will come up on a certain roll is  ALWAYS 1/6.

7. What is the probability of rolling 7 OR 11?  
2/9
or 0.2222
  6/36 = 1/6 probability of rolling 7 
 2/36=1/18 probability of rolling 11
the probability of rolling 7 OR 11 will be:
1/6 + 1/18= 3/18 +1/18 = 4/18= 2/9
8. What is the probability of rolling 7 AND then 11? 
1/108 or
0.009259
 the chance of rolling a 7 is 1/6
the chance of rolling a 11 is 1/18
the chance of rolling a 7 AND 11 is 1/6*1/18 = 1/108
0.009259
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  
 .4914
 The probability of rolling a 12 is 1/36 =0.027
Consequently, the probability of not rolling a 12 is 35/36, and in n trials will be
 (35/36)^n.
So the probability of rolling at least one 12 in n trials is 1-(35/36)^n.
Then we have:
if n=24
1-(35/36)^24= 1- (0.97)^24= 1-0.50859612386909674041792802317136= 0.491403876130903259582071976829=  .4914


From wHolt - 11/21/06 1:02 PM

TBird - thanks for going first
you forgot 6,1 and some others

From TBird - 11/20/06 10:03 PM

QUESTIONSANSWERHOW?
1. How many ways can you roll two 6-sided dice?   21 ways

  1,1  2,2  3,3  4,4  5,5  6,6 
  1,2  2,3  3,4  4,5  5,6

   1,3  2,4  3,5  4,6

   1,4  2,5  3,6

  1,5  2,6

  1,6

2. How many ways can you roll a 7?
(How many ways can the spots add up to 7?)  		 3 ways 1,6  2,5  3,4
3. What is the probability of rolling a 7?   14.28%  3/21 = 0.142857 = 14.28%
4. What is the probability of NOT rolling a 7?  85.71% 18/21 = 0.857142  = 85.71%
5. What is the sum of the last two probabilities?  0.999999= 99.99% or 100% 

 85.71 + 14.28 = 0.999999= 99.99% or 100% 
6. If you roll a 7 nine times in a row,
what is the probablity you will roll 7 on your tenth roll?  		 14%
 The probability will always stay the same it will be 3/21 or 0.142857 which equals 14%

7. What is the probability of rolling 7 OR 11?  

 19.04% 4/21 = 0.190476 = 19.04%

8. What is the probability of rolling 7 AND then 11? 

  
9. If you roll two dice 24 times,
what is the probability AT LEAST one roll will be a 12?  		 24/21 + 1 = 2.142857 2.142857/21 = 0.102040 = 10.2%

10. If you roll 4 dice,
what is the probability AT LEAST one die will be a 6?

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Last Modified 11/13/06 10:10 AM

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