Probability Of Six Sided Dice

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Probability of Six Sided Dice:

ROLL OF DICE IN PROBABILITY: How dice are used for statistics, and probability that is discussed here. Dice have a major role in probability and problems involving probability. For a single roll of common dice with 6 faces, Probability of getting any number from 1 to 6 = 1/6 If the die has say n faces, then probability of getting one face =1/n Hence throwing die forms an example of discrete uniform distribution when a single die is thrown.

Consider the case of throwing two dice together: Sum 2 3 4 5 6 7 8 9 10 11 12 Prob 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/46 2/36 1/36 Because getting 2 and 12 has one change 1 or 6 on both dice while getting 7 has maximum change as 7 = 1,6 or 2,5,or 3,4 or 4,3 or 5,2 or 6,1 So it uniformly increases goes to maximum then again comes down. For more dice, if we draw a curve, it will be more bell shaped going peak in the middle again coming gradually down uniformly symmetrical both sides of the maximum. Hence the exact probability distribution F s,n of a sum of n sided dice can be calculated as the repeated convolutio of the single-die probability ...
... distribution with itself. For example here F s,n (k) = sigma F s,1(i) F s,n-1(k-i) for i=1 to k-n+1 The explanation for the formula is we want total number to be k. Sum of all faces = k. That different ways are expressed in sigma with notations. We can substitute for any integer k and get the answer for all n using this formula. Application: In the above we got for k=7, when two dice were rolled is 6/36 Using the formula also we get the same result as Prob (getting 7) = F = P(1)P(6) + P(6)P(1)+P(2)P(5)+P(5)P(2)+....=6/36 Same answer. When n increases this formula only can be used as practically writing all combinations and counting becomes impossible and time consuming.

Rolling of Two 6 Sides Dice:

DICE PROBABILITIES THE STATISTICAL OUTCOMES OF ROLLING 2 SIX SIDED DICE Rolling two six sided dice is common in most popular board games like Monopoly, Backgammon, and settlers of Catan. There are actually some charts giving percentages for each outcome Here below is an example for 2 dice: 2--2.78% 4--8.33% 6--13.89% 8--13.89% 10--8.33% 12--2.78% 3--5.56% 5--11.11% 7--16.67% 9--11.11% 11--5.56% Here 7 has maximum chance. Hence gamblers can select 7 for their increased chance of winning. Like this for any number of dice, percentage chart is available, giving us a hint which has maximum chance.

Maximum Number of Faces in a Dice:

Though there is no restriction on number of faces, but for the limited dimension we use for playing, 20 is the ideal maximum no of face, because more than 20 faces may make the die circular and it will be rolling without stopping or even if it stops more than one face will be facing up. Hence 20 is consider optimum size for common dice. ILLUSTRATION ON DICE PROBLEM IN STATISTICS: In a game three dice are thrown simultaneously. Find the probability of sum of maximum of 17. Solution: Here the required probability is sum of getting maximum 17. Reqd prob = prob(getting 3,4,5,6....17) = 1 - prob (18) = 1-1/6^3 =1-1/216 =215/216

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