Multiple Dice to Produce a Randomized Result from a Predetermined Probability Distribution

Abstract

One embodiment of a plurality of dice with markings such that when the primary die is rolled a user either lands on a randomized result within the predetermined probability distribution or is directed to roll a secondary die which either provides a result or directs the user to roll a tertiary die. In the embodiments discussed here the final result is achieved on the first roll over 90% of the time. The embodiments described and shown are capable of representing complex distributions that include highly improbable outcomes, providing users with the convenient alternative of simply reading the result rather than performing comparative and mathematical operations on the results of multiple dice rolls.

Description

TECHNICAL FIELD

The present invention relates to chance devices, more specifically, dice.

LIST OF PRIOR ART

The following is a tabulation of some prior at that presently appears relevant:

	U.S Patent Number	Issued	Patentee
	3,208,754	September 1965	Sieve
	5,556,096	September 1996	Eardley et al.
	6,926,276	August 2005	Zocchi

NONPATENT LITERATURE DOCUMENTS

• Roberts, Siobhan, The New Yorker, “The Dice You Never Knew You Needed” (Apr. 26, 2016) https://www.newyorker.com/tech/elements/the-dice-you-never-knew-you-needed
• thedicelab.com/D60Uses.html
• thedicelab.com/d120.html

BACKGROUND—PRIOR ART

Tabletop games have used dice as a method for introducing random chance for hundreds of years. However, as the popularity of tabletop role-playing games has increased over the past decades, so has the complexity of the distributions that these dice seek to represent. One answer was to increase the variety of the dice employed, for example U.S. Pat. No. 3,208,754 to Sieve (1965) included dice with more sides than a standard cube, such as a 12 and 20-sided die. However, even these dice used in isolation could not keep up with the demand to represent complex distributions.

Games and players created different operations using multiple dice in combination. To provide a player with a boosted chance at a high roll the player could roll the die twice and take the higher result Conversely, a player could have their expected result lowered by rolling a die twice and taking the lower result. These two operations became known as rolling with advantage and disadvantage respectively. Even in the most popular of tabletop RPGs a new player must roll a die 24 times, summing up various values that meet certain criteria just to create their character. Very advanced players can find themselves having to roll a die up to 40 times in certain circumstances.

Dice with more faces were introduced but met with a number of problems. A die of a given diameter becomes increasingly rounded as the number of faces is increased, causing the die to roll perpetually and making it difficult for the player to determine which face is up when it finally comes to rest. U.S. Pat. No. 6,926,276 to Zocchi (2005) described a braking system to be used to slow such dice, but some players were put off by the jerking motion that resulted and even a perfect braking system wouldn't help players determine which face was up on an overly rounded die.

One manufacturer eventually introduced 60-sided and 120-sided dice. The dice were much larger than average dice in an attempt to fix the roundness issue. However, a new complication arose. The existing games had no specified use for these larger dice and even a 120-sided die can't represent the complexity of operations such as rolling a 20-sided die twice, which creates 400 possible outcomes. The creator of that 120-sided die admitted that they had no idea what to use the die for. Yet, even without a specific use the manufacturer pre-sold over a thousand of the dice. Players so badly wanted to use larger dice with more faces that they came up with complex conversion charts and creative attempts to incorporate them into their games, or simply added the larger dice to their collections as an item for display and starting conversation.

Despite the various attempts at improving dice with many faces over the years, a number of known disadvantages still remain:

• • a) They can't reflect complex distributions created by games and players. Even relatively simple operations like rolling with advantage have improbable outcomes that would require hundreds of faces to represent Even though such dice could be created by sidestepping the geometric limitations using the facets on a sphere technique as seen in U.S. Pat. No. 5,556,096, Eardley et al (1996), they would have to be so large as to become unusable in most circumstances.
  • b) They can't be used in popular tabletop RPGs without some sort of conversion system or secondary mathematical operation
  • c) They are too round. Even larger versions of the dice suffer from the roundness problem as a result of the manufacturing process used for the vast majority of dice. Most often, dice are injection molded in a mold that has both the numbers and the geometry, covered entirely in paint, and the excess paint not sitting within the number grooves is removed through abrasion by tumbling or vibration. This process results in a die with rounded over edges, dramatically decreasing the effective size of the faces and hindering their ability to stop the rolling die or clearly display a result.

SUMMARY

A method for producing a randomized result within a predetermined probability distribution comprising rolling a primary die and rolling one or more additional dice if directed.

DRAWINGS—FIGURES

FIG. 1 shows an overview of the dice that comprise embodiment one

FIG. 2 shows a flattened view of the primary die in embodiment one

FIG. 3 shows a flattened view of the secondary die in embodiment one

FIG. 4 shows a flattened view of the tertiary die in embodiment one

FIG. 5 details the D20 With Advantage distribution

FIG. 6 details the distribution of results for embodiment one

FIG. 7 shows an overview of the dice that comprise embodiment two

FIG. 8 shows a flattened view of the primary die in embodiment two

FIG. 9 shows a flattened view of the secondary die in embodiment two

FIG. 10 shows a flattened view of the tertiary die in embodiment two

FIG. 11 details the D20 With Disadvantage distribution

FIG. 12 details the distribution of results for embodiment two

FIG. 13 shows an overview of the dice that comprise embodiment three

FIG. 14 shows a flattened view of the primary die in embodiment three

FIG. 15 shows a flattened view of the secondary die in embodiment three

FIG. 16 details the distribution of results for embodiment three

FIG. 17 compares the distribution for embodiment three to the D20 With Advantage distribution

FIG. 18 shows an overview of the dice that comprise embodiment four

FIG. 19 shows a flattened view of the primary die in embodiment four

FIG. 20 shows a flattened view of the secondary die in embodiment four

FIG. 21 details the distribution of results for embodiment four

FIG. 22 compares the distribution for embodiment four to the D20 With Disadvantage distribution

FIG. 23 shows an overview of the dice that comprise embodiment five

FIG. 24 shows a flattened view of the primary die in embodiment five

FIG. 25 shows a flattened view of the secondary die in embodiment five

FIG. 26 details the 4D6 Drop Lowest distribution

FIG. 27 details the distribution of results for embodiment five

FIG. 28 compares the distribution for embodiment five to the 4d6 Drop Lowest distribution

Drawings - Reference Numerals
1  Embodiment one primary die
2  Embodiment one secondary die
3  Embodiment one tertiary die
4  Embodiment two primary die
5  Embodiment two secondary die
6  Embodiment two tertiary die
7  Embodiment three primary die
8  Embodiment three secondary die
9  Embodiment four primary die
10 Embodiment four secondary die
11 Embodiment five primary die
12 Embodiment five secondary die

DETAILED DESCRIPTION—ADVANTAGES

Accordingly several advantages of one or more aspects are as follows: to provide a system of dice

• • a) capable of producing a randomized result within a probability distribution that contains outcomes too improbable to be shown on existing dice,
  • b) that allows for the final result to simply be read off the die without requiring conversion systems or secondary mathematical operations,
  • c) that produces the final result in a single roll over 90% of the time,
  • d) that allows the player an opportunity to use larger dice types not currently widely used in play,
  • e) that employs unique numbering schemas that not only give the dice a novel appearance but also serve the practical purpose of showing a visual representation of the probability of each result,
  • f) that uses dice precisely machined with well-defined edges that allow the die to stop well and clearly display the result despite having many faces.

Other advantages will be apparent from a consideration of the drawings and ensuing description.

Detailed Description—Materials, Finishes, Markings, and Geometry

All of the dice in every embodiment are machined from solid aluminum, anodized, and marked with a laser in a contrasting lighter mark. All of the primary dice (1, 4, 7, 9, 11) have a diameter of 53 mm as measured from face to opposite face and the geometry of a disdyakis triacontahedron. The color of the anodize is a charcoal in each embodiment relating to the D20 with Advantage (embodiments one and three) and black in each embodiment relating to the D20 with Disadvantage (embodiments two and four). The color coding for embodiments one through four serve to help a player who owns both an advantage and disadvantage embodiment to easily tell them apart The anodize of embodiment five is black. All of the 60-sided secondary dice (2, 3, 5, 6, 8, 10) as seen in embodiments one, two, three, and four measure 37 mm from face to opposite face. However, embodiments one and two use the pentagonal hexecontahedron geometry while embodiments three and four use the geometry of the pentakis dodecahedron. The secondary die 12 (FIGS. 23, 25) in embodiment five is a cube that measures 17 mm from face to opposite face.

Detailed Description—Embodiment One—FIGS. 1-4, 6

Embodiment one is illustrated in FIG. 1 and is comprised of primary die 1, secondary die 2, and tertiary die 3. The markings on the primary faces are illustrated in the flattened view of the primary die 1 in FIG. 2. These primary markings consist of numerals from 3-20 and the marking “R3”. The secondary markings on die 2 in FIG. 3 consist of numerals that range from 1-20 and the marking “R4”. The numeral 3 is encircled. The tertiary markings on die 3 in FIG. 4 consist of numerals that range from 1-20 with the numeral 4 encircled. FIG. 6 details all of the markings on the dice in this embodiment with the primary marking “R3” listed as “Roll Secondary Die” and the secondary marking “R4” listed as “Roll Tertiary Die”.

Operation—Embodiment One—FIGS. 1-6

Embodiment One is used to replace the operation of rolling a D20 (a die numbered 1-20 where each result has a 5% chance of occurring) twice and taking the higher value. This operation is referred to as rolling a D20 with advantage. FIG. 5 details the D20 With Advantage probability distribution.

A user begins by rolling the primary die 1 (FIGS. 1, 2). 91.67% of the time the primary die will show a final numerical result. In this most common case, the user has completed their roll with advantage while only having to roll the die once and simply reading the result 8.33% of the time the user will roll a value of “R3”, which indicates to the user that they must roll the secondary die 2 (FIGS. 1,3). The marking “R3” is used because the secondary die 2 (FIGS. 1, 3) has the numeral 3 on some faces while the tertiary die 3 (FIGS. 1, 4) does not To assist the user in identifying the secondary die 2 (FIGS. 1, 3), each numeral 3 is encircled.

Similarly to the primary markings (FIG. 2), the secondary markings (FIG. 3) include final result numerals and some faces marked “R4” which direct the user to roll the tertiary die 3 (FIGS. 1, 4). The marking “R4” is used because the tertiary die 3 (FIGS. 1, 4) has the numeral 4 on some faces and the secondary die 2 (FIGS. 1, 3) does not To assist the user in identifying the tertiary die 3 (FIGS. 1, 4), each numeral 4 is encircled.

The user will reach a final result on either the primary or secondary die 98.33% of the time. All of the tertiary faces (FIG. 4) are marked with a final result, so in a worst case scenario the user would have to roll all three dice to get their final result. However, even in that case they can simply read the result from the die without the hassle of remembering or comparing values from multiple rolls.

FIG. 6 shows the probability distribution that results from using the three dice in embodiment one and demonstrates that the distribution is identical to the D20 With Advantage distribution shown in FIG. 5. The first section in FIG. 6 is labeled “Primary Die” and details all the markings on the primary die 1 (FIGS. 1,2). 8.333% (10/120) of primary faces indicate that the user must roll the secondary die 2 (FIGS. 1, 3). Thus, the remaining probability of 8.333% is carried over to the secondary die 2 (FIGS. 1, 3) and divided across the 60 secondary faces, each of which has a probability of 0.139%. The 1.667% probability of the secondary die landing on“Roll Tertiary Die” (indicated by the secondary marking “R4” as seen in FIG. 3) is similarly carried over to the tertiary die 3 (FIGS. 1, 4) and divided among its 60 faces, each of which represents a 0.028% probability.

The “Combined” section to the right sums all of the final results (the results numbered 1-20, not counting any intermediary result that directs the user to roll an additional die) and shows them to be identical to the D20 With Advantage distribution in FIG. 5. While many of the values shown in FIG. 6 were rounded to the decimal places shown, all calculations were done with exact numbers and the probability distribution of the dice used as described in this embodiment is mathematically identical to the D20 With Advantage distribution.

Detailed Description—Embodiment Two—FIGS. 7-10, 12

Embodiment Two is illustrated in FIG. 7 and is comprised of primary die 4, secondary die 5, and tertiary die 6. The markings on the primary faces are illustrated in the flattened view of the primary die 4 in FIG. 8. These primary markings consist of numerals from 1-18 and the marking “R3”. The secondary markings on die 5 in FIG. 9 consist of numerals ranging from 1-20 and the marking “R5”. The numeral 3 is encircled. The tertiary markings on die 6 in FIG. 10 consist of numerals from 1-20 with the numeral 5 encircled. FIG. 12 details all of the markings on the dice in this embodiment with the primary marking “R3” listed as “Roll Secondary Die” and the secondary marking “R5” listed as “Roll Tertiary Die”.

Operation—Embodiment Two—FIGS. 7-12

Embodiment Two is used to replace the operation of rolling a D20 twice and taking the lesser value, which is to say rolling a D20 with disadvantage. FIG. 11 details the D20 With Disadvantage probability distribution that results from rolling a D20 with disadvantage as described above.

A user begins by rolling the primary die 4 (FIGS. 7, 8). 91.67% of the time the initial roll of the primary die will show a final result which is equivalent to the user having to roll a D20 twice and choose the lesser value. 8.33% of the time the user will roll a value of “R3”, indicating that the user must roll the secondary die 5 (FIGS. 7, 9). The marking “R3” is used because the secondary die 5 (FIGS. 7, 9) has the numeral 3 on some faces while the tertiary die 6 (FIGS. 7, 10) does not. To assist the user in identifying the secondary die 5 (FIGS. 7, 9), each numeral 3 is encircled.

If the result of the primary die 4 (FIGS. 7, 8) roll is “R3” the user will roll the secondary die 5 (FIGS. 7, 9). Similarly to the primary markings (FIG. 8), the secondary markings (FIG. 9) include final result numerals and some faces marked “R5” which directs the user to roll the tertiary die 6 (FIGS. 7, 10). The marking “R5” is used because the tertiary die 6 (FIGS. 7, 10) has the numeral 5 on some faces and the secondary die 5 (FIGS. 7, 9) does not To assist the user in identifying the tertiary die 6 (FIGS. 7, 10), each numeral 5 is encircled.

The user will reach a final result on either the primary or secondary die 98.33% of the time, usually on the first roll. Rarely will the tertiary die 6 (FIGS. 7, 10) be required.

FIG. 12 shows the probability distribution that results from using the three dice in Embodiment Two and demonstrates that the distribution is identical to the D20 With Disadvantage distribution shown in FIG. 11. The first section in FIG. 12 is labeled “Primary Die” and details all the markings on the primary die 4 (FIGS. 7, 8). 8.333% (10/120) of primary faces indicate that the user must roll the secondary die 5 (FIGS. 7, 9). Thus, the remaining probability of 8.333% is carried over to the secondary die 5 (FIGS. 7, 9) and divided across the 60 secondary faces, each of which has a probability of 0.139%. The 1.667% probability of the secondary die landing on“Roll Tertiary Die” (indicated by the secondary marking “R5” as seen in FIG. 9) is similarly carried over to the tertiary die 6 (FIGS. 7, 10) and divided among its 60 faces, each of which represents a 0.028% probability.

The “Combined” section to the right sums all of the final results and shows them to be identical to the D20 With Disadvantage distribution in FIG. 11. While many of the values shown in FIG. 12 were rounded to the decimal places shown, all calculations were done with exact numbers and the probability distribution of the dice used as described in this embodiment is mathematically identical to the D20 With Disadvantage distribution.

Detailed Description—Embodiment Three—FIGS. 13-16

Embodiment Three is illustrated in FIG. 13 and is comprised of primary die 7 and secondary die 8. The markings on the primary faces are illustrated in the flattened view of the primary die 7 in FIG. 14. These primary markings consist of numerals in the range of 2-20 and two markings in the shape of an arrow. The secondary markings on die 8 in FIG. 15 consist of numerals in the range of 1-20. FIG. 16 details all of the markings on the dice in this embodiment with the arrow markings listed as “Roll Secondary Die”.

Operation—Embodiment Three—FIGS. 5, 13-17

Embodiment Three is used to replace the operation of rolling a D20 with advantage. FIG. 5 details the D20 With Advantage probability distribution. All final results in this embodiment belong to a probability distribution that is a close approximation to the D20 With Advantage distribution. In this case the approximation allows for a less complex group of dice and a better experience for the user. In tabletop RPGs, extreme precision of probabilities is much less important than creating an enjoyable shared experience for the players. In this context, an approximation that is very nearly aligned with the exact D20 With Advantage distribution would be permissible to the vast majority of players.

A user begins by rolling the primary die 7 (FIGS. 13, 14). 98.33% of the time the primary die will show a final result 1.67% of the time the user will land on the arrow marking which indicates to the user that they must roll the secondary die 8 (FIGS. 13, 15) to get their final result. In this embodiment, a user must roll a maximum of twice, but almost always would get their result in a single roll. This is a very pleasant experience for the user. Furthermore, rolling with advantage is exciting in itself, so having a special large die to use enhances that experience.

FIG. 16 shows the probability distribution that results from using the two dice in Embodiment Three and demonstrates that the distribution is an approximation of the D20 With Advantage distribution shown in FIG. 5. FIG. 16 is read in the same way as FIGS. 6 and 12.

FIG. 17 displays the D20 With Advantage distribution compared to the distribution that is achieved using the dice of embodiment three. While probabilities of each result are very similar when the two distributions are compared, the results of 1, 4, 19, and 20 are identical to the D20 With Advantage distribution and the mean of the two distributions is also equal. This is relevant, since in many tabletop RPGs the “critical” rolls of 1 and 20 are treated differently than other rolls, so it is preferable to replicate the exact probabilities in an approximation such as this. Furthermore, in some circumstances a 19 is also considered a critical roll.

Detailed Description—Embodiment Four—FIGS. 18-22

Embodiment Four is illustrated in FIG. 18 and is comprised of primary die 9 and secondary die 10. The markings on the primary faces are illustrated in the flattened view of the primary die 9 in FIG. 19. These primary markings consist of numerals in the range of 1-19 and two markings in the shape of an arrow. The secondary markings on die 10 in FIG. 20 consist of numerals in the range of 1-20. FIG. 21 details all of the markings on the dice in this embodiment with the arrow markings listed as “Roll Secondary Die”.

Operation—Embodiment Four—FIGS. 11, 18-22

Embodiment Four is used to replace the operation of rolling a D20 with disadvantage. All final results in this embodiment belong to a probability distribution that is a close approximation to the D20 With Disadvantage distribution as detailed in FIG. 11.

A user begins by rolling the primary die 9 (FIGS. 18, 19). The initial roll will show a final result 98.33% of the time. Only in 1.67% of the cases will the user roll the secondary die 10 (FIGS. 18, 20).

FIG. 21 shows the probability distribution resulting from using the two dice in Embodiment Four and demonstrates that the it is an approximation of the D20 With Disadvantage distribution shown in FIG. 11.

FIG. 22 displays the D20 With Disadvantage distribution compared to the distribution that is achieved using the dice of embodiment four. While probabilities of each result are very similar when the two distributions are compared, the results of 1, 2, 17, and 20 are identical to the D20 With Disadvantage distribution and the mean of the two distributions is also equal.

In both embodiment three and four the secondary die has been designed to create a feeling of suspense in the player as its markings are concentrated in the highest and lowest values. In embodiment four, landing on the arrow marking on the primary die is a chance at redemption, since there is a 43.33% chance to roll a 17 or higher. Conversely in embodiment three the arrow marker is perilous. While there is a 56.67% chance of an 18 or above (FIG. 16), there alternative is a 4 or below which is a disastrous outcome for a roll with advantage. These numbering schemas where the most extreme values are placed on the secondary die create an opportunity for the player to experience a feeling of nervous excitement. As a result, the secondary die becomes a fun addition rather than just another die used to resolve the distribution.

Detailed Description—Embodiment Five—FIGS. 23-25, 28

Embodiment Five is illustrated in FIG. 23 and is comprised of primary die 11 and secondary die 12. The markings on the primary faces are illustrated in the flattened view of the primary die 11 in FIG. 24. These primary markings consist of numerals in the range of 5-18 and one marking in the shape of an arrow. The secondary markings on die 12 in FIG. 25 consist of numerals in the range of 1-3. FIG. 27 details all of the markings on the dice in this embodiment with the arrow marking listed as “Roll Secondary Die”.

Operation—Embodiment Five—FIGS. 23-28

Embodiment Five is used to replace the operation of rolling a D6 (a die with possible results 1-6 each with a 1 in 6 chance of occurring) four times, summing the four results, and subtracting the lowest of the four values. This operation is commonly undertaken by tabletop RPG players to determine the ability scores of a new character. FIG. 26 details the probability distribution that is created by this operation, which I will refer to as the 4D6 Drop Lowest distribution. All final results in this embodiment belong to a probability distribution that is an approximation of the 4D6 Drop Lowest distribution.

A user begins by rolling the primary die 11 (FIGS. 23, 24). 99.167% of the time the primary die will show a final result 0.833% of the time the user will land on the arrow marking which indicates to the user that they must roll the secondary die 12 (FIGS. 23, 25). All outcomes of rolling the secondary die 12 (FIGS. 23, 25) are final outcomes.

FIG. 27 shows the probability distribution that results from using the two dice in embodiment five and demonstrates that the distribution is an approximation of the 4D6 Drop Lowest distribution shown in FIG. 11.

The “Combined” section to the right sums all of the final results and shows them to be an approximation of the 4D6 Drop Lowest distribution in FIG. 26. FIG. 28 displays the 4D6 Drop Lowest distribution compared to the distribution that is achieved using the dice of embodiment five. The probabilities of each result are very similar when the two distributions are compared and the means of the two distributions are within 0.01 of each other.

This was a difficult distribution to approximate with some very improbable outcomes. However, the embodiment shown here is extremely helpful to new players and those who wish to introduce others to the game. In the current most popular tabletop RPG the 4D6 Drop Lowest operation must be undertaken six times in the process of rolling a new character. Imagine an experienced player guiding five new players through the process of creating their first character. Instead of 120 rolls that each have to be remembered, compared and summed by novice players, each new player would have to roll the large, heavy primary die just six times and simply read the values. The odds are that perhaps one player may have to use the secondary die, but it is very possible the secondary die would never be used. This embodiment provides a useful tool for new players and experienced game masters alike.

CONCLUSION, RAMIFICATIONS, AND SCOPE

The dimensions, geometries, materials, markings, and manufacturing techniques described above and in the following embodiments are what I presently contemplate for each embodiment but other dimensions, geometries, materials, markings, numbering schemas, and manufacturing techniques could be used. For instance, the dice could be made either larger or smaller and from materials such as woods, metals, stones, polymers, fossils, amber, bone, etc. Instead of machining the dice, dice made of suitable materials could be molded or cast using methods that do not compromise the precision geometry and crisp edges. If the molds contain markings then no laser markings would be required. Markings could also be engraved onto the dice or applied with paint or ink rather than marked with a laser.

Additional embodiments could include:

• • a) Representing other useful distributions. For instance, many damage rolls in tabletop RPGs require many dice. Each of these distributions could have its own embodiment, as could any other useful distribution.
  • b) Using more rounds of dice than primary, secondary and tertiary to represent even more complex distributions. This technique also enables the use of more, smaller dice, which could be useful in creating a physically smaller version of the system, for instance one intended as a travel version.
  • c) Using rounds that have more than one die in each. For instance, a very complex distribution can be resolved in only two rounds by using a primary die that designates multiple secondary dice, ensuring that the distribution resolves in a maximum of two rolls.
  • d) Combining the above methods with the many different dice that exist; most commonly D4, D6, D8, D10, D12, D20, more rarely D2, D3, D5, D7, D14, D16, D18, D24, D30, D34, D48, D50, D60, D120, as well as the infinite types of dice that can be made using the facets on a sphere technique (as seen in U.S. Pat. No. 5,556,096, Eardley et al., September 1996) such as the D100.
  • e) Using the dice with the same number of faces described here but with different numbering schemas which still achieve the same distribution.

Even extremely complex distributions can be represented in a way where the user can simply read the result off the die and come to a solution in a single roll a majority of the time. All of the potentially infinite number of alternative solutions for any distribution rely on the core principle of using sequential rolls of dice that use non-standard numbering schema and compound probabilities to reproduce a predetermined distribution.

Accordingly, the scope should be determined not by the embodiments illustrated, but the appended claims and their legal equivalents

Claims

1. A method for producing a randomized result within a predetermined probability distribution comprising: a. rolling a primary die with means for indicating said randomized result and means for directing a user to roll a secondary die, andb. following any direction given by said primary die to roll said secondary die with means for indicating said randomized result,

2. The method from claim 1 wherein said primary die has an insufficient quantity of faces to produce said randomized result when used in isolation because said predetermined probability distribution includes results which are less probable than the probability of rolling any given face on said primary die.

3. The method from claim 1 wherein said primary die is selected from the group consisting of 120-sided and 60-sided dice.

4. The method from claim 1 wherein said secondary die is selected from the group consisting of 120-sided, 100-sided, 60-sided, 50-sided, 48-sided, 30-sided, 24-sided, 20-sided, 12-sided, 10-sided, 8-sided, 6-sided and 4-sided dice.

5. The method from claim 1 wherein said secondary die has means for directing said user to roll a tertiary die and when said primary die, said secondary die, and said tertiary die are used in combination they produce said randomized result.

6. The method from claim 1 wherein said predetermined probability distribution is the D20 With Advantage probability distribution that results from rolling two 20-sided dice each numbered 1-20 and choosing the greater value as the result.

7. The method from claim 1 wherein said predetermined probability distribution is an approximation of the D20 With Advantage probability distribution that results from rolling two 20-sided dice each numbered 1-20 and choosing the greater value as the result.

8. The method from claim 1 wherein said predetermined probability distribution is the D20 With Disadvantage probability distribution that results from rolling two 20-sided dice each numbered 1-20 and choosing the lesser value as the result.

9. The method from claim 1 wherein said predetermined probability distribution is an approximation of the D20 With Disadvantage probability distribution that results from rolling two 20-sided dice each numbered 1-20 and choosing the lesser value as the result.

10. The method from claim 1 wherein said predetermined probability distribution is an approximation of the 4D6 Drop Lowest probability distribution that results from rolling four 6-sided dice each numbered 1-6 and summing the three greatest values to come to the result.

11. A plurality of dice for producing a randomized result conforming to a predetermined probability distribution, comprising: a. a primary die with a predetermined number of primary faces, andb. a secondary die with a predetermined number of secondary faces, andc. a plurality of primary markings that appear on said primary faces and are selected from the group consisting of result markings and markings directing a user to roll said secondary die, andd. a plurality of secondary markings that appear on said secondary faces and indicate a result,

12. The plurality of dice from claim 11 wherein said primary die has an insufficient quantity of said primary faces to produce said randomized result when used in isolation because said predetermined probability distribution includes results which are less probable than the probability of rolling any individual example of said primary faces.

13. The plurality of dice from claim 11 wherein said predetermined number of primary faces is selected from the group consisting of 120 and 60.

14. The plurality of dice from claim 11 wherein said predetermined number of secondary faces is selected from the group consisting of 120, 100, 60, 50, 48, 30, 24, 20, 12, 10, 8, 6 and 4.

15. The plurality of dice from claim 11 wherein said plurality of dice includes a tertiary die and said plurality of secondary markings includes markings directing said user to roll said tertiary die and when said user starts by rolling said primary die and follows any potential directions given by said primary die to roll said secondary die and follows any potential directions given by said secondary die to roll said tertiary die, said randomized result is eventually produced by a member of said plurality of dice.

16. The plurality of dice from claim 11 wherein said predetermined probability distribution is the D20 With Advantage probability distribution that results from rolling two 20-sided dice each numbered 1-20 and choosing the greater value as the result.

17. The plurality of dice from claim 11 wherein said predetermined probability distribution is an approximation of the D20 With Advantage probability distribution that results from rolling two 20-sided dice each numbered 1-20 and choosing the greater value as the result.

18. The plurality of dice from claim 11 wherein said predetermined probability distribution is the D20 With Disadvantage probability distribution that results from rolling two 20-sided dice each numbered 1-20 and choosing the lesser value as the result.

19. The plurality of dice from claim 11 wherein said predetermined probability distribution is an approximation of the D20 With Disadvantage probability distribution that results from rolling two 20-sided dice each numbered 1-20 and choosing the lesser value as the result.

20. The plurality of dice from claim 11 wherein said predetermined probability distribution is an approximation of the 4D6 Drop Lowest probability distribution that results from rolling four 6-sided dice each numbered 1-6 and summing the three greatest values to come to the result.