Lesson Tossing a Die or Dice
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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 . 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 . {{{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))}}} . 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? . . . . (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? . . . . 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