Lesson Tossing a Die or Dice

Source code of 'Tossing a Die or Dice'

This Lesson (Tossing a Die or Dice) was created by by Nate(3500)

Dice:
A die generally has six sides. To determine the amount of possible outcomes, take 6 to the power of the amount of rolls.
Your ODDS of getting a number you want is defined as: (success)/(failure)
If you wanted one six tossing one die:
success: 6 ~> one chance
failure: 1, 2, 3, 4, 5 ~> five chances
1:5 or 1/5
Remember, this is not your probability. This is yours ODDS. Probability is defined as: (success)/(success + failure)
If you wanted one six tossing one die:
success: 6 ~> one chance
success + failure: 1, 2, 3, 4, 5, 6 ~> six chances
1/6 or about 0.1666 or about 17%
Now, find the ODDS of getting a number less than four tossing one die:
success: 1, 2, 3 ~> three chances
failure: 4, 5, 6 ~> three chances
3:3 or 3/3 *You can not take 3/3 to equal 1 because you do not have a 100% of getting what you want.
Tossing a die twice: 6^2 = 36 possible outcomes
What is the probability of getting a six in a row?
Success: (6,6) there is only one possible way of getting a six in a row
Success + Failure: 36
1/36 or about 0.0277 or about 3%
Another way:
Chances of getting a six ~> (1/6)
Chances of getting another six ~> (1/6)
(1/6)(1/6) = 1/36
What is the probability of getting a number twice?
Chances of getting a number ~> (6/6)
Chances of getting that number ~> (1/6)
(6/6)(1/6) = 1/6
What is the probability of getting a toss of two dice with the sum of 3?
Number of possible outcomes = 6^2 = 36
Success: (1,2) and (2,1) ~> two chances
2/36 = 1/18 or or about 0.055555 or about 6%
Here is an easy made list for tossing two dice:
SUM[2] = 1
SUM[3] = 2
SUM[4] = 3
SUM[5] = 4
SUM[6] = 5
SUM[7] = 6
SUM[8] = 5
SUM[9] = 4
SUM[10] = 3
SUM[11] = 2
SUM[12] = 1
Notice that the best possible sum is: (highest sum + lowest sum)/2
Rolling two die: (12 + 2)/2 = 14/2 = 7 The best sum is seven.
This is featured as: f(x) = -|x - 7| + 6 where {{{x}}} is the sum of the dice and {{{y}}} is the chances of getting that sum.
{{{graph(500,400,-2,16,-2,8,-sqrt((x - 7)^2) + 6)}}}
The number of possible outcomes tossing three dice: 6^3 = 216
What is the probability of getting a sum of 5?
Success: (1,1,3) ; (1,3,1) ; (3,1,1) ; (2,2,1) ; (2,1,2) ; (1,2,2) ~> 6 chances
6/216 = 1/36 or about 0.027777 or about 3%
Best chances of getting a sum: (18 + 3)/2 = 21/2 = 10.5
Since you can not get a sum of 10.5, you would choose the integers on both sides. The best chances would be either a sum of 10 or a sum of 11.
Here is an easy made list for tossing three dice:
SUM[3] = 1
SUM[4] = 3
SUM[5] = 6
SUM[6] = 10
SUM[7] = 15
SUM[8] = 21
SUM[9] = 28
SUM[10] = 36
SUM[11] = 36
SUM[12] = 28
SUM[13] = 21
SUM[14] = 15
SUM[15] = 10
SUM[16] = 6
SUM[17] = 3
SUM[18] = 1
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Equation: f(x) = 0.5( (10.5 - |x - 10.5|)^2 - 3(10.5 - |x - 10.5|) + 2)
~ where x is the sum of die
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{{{graph(400,500,-1,20,-1,40,0.5( (10.5 - sqrt((x - 10.5)^2) )^2 - 3(10.5 - sqrt((x - 10.5)^2)) + 2))}}}
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Chances of getting a sum of 11 by tossing three dice?
36/216 = 1/6 or about 17%
*By reading what you did, try solving these:
What is the chances of getting a Yahtzee on the first roll?
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(6/6)(1/6)(1/6)(1/6)(1/6) = 1/1296 or about 0.0007716049 or about 0.077%
What is the chances of getting a sum of 6 by tossing two dice?
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success: 5 chances
total possibilities: 36
5/36 or about 0.1388 or about 14%
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
x(x + 1)/2
f(x) = -|x - 10.5| + x(x + 1)/2