Dice

Names

 * d4 (tetrahedron): The only common die that cannot actually roll or land with one side up. Instead, you usually toss it in the air and after it lands, read the number written either near the vertex or near the face opposite to it, depending on design.
 * d6 (cube): Easily the most common type, often synonymous with the word "die" itself.
 * d8 (octahedron)
 * d10 (pentagonal trapezohedron): The only common die that isn't a Platonic solid.
 * d12 (dodecahedron)
 * d20 (icosahedron)

Ref: http://tvtropes.org/pmwiki/pmwiki.php/UsefulNotes/Dice

Probability
Probability of an event happening = Number of ways it can happen / Total number of outcomes

Two Dice Probability
Simple Probability: 2	 3% 3	 6% 4	 8% 5	11% 6	14% 7	17% 8	14% 9	11% 10	 8% 11	 6% 12	 3%

Probability: 2	 2.78% 3	 5.56% 4	 8.33% 5	11.11% 6	13.89% 7	16.67% 8	13.89% 9	11.11% 10	 8.33% 11	 5.56% 12	 2.78%

Calculated: d1	d2	sum		num	count	prob 1	1	2		2	1	2.78% 1	2	3		3	2	5.56% 1	3	4		4	3	8.33% 1	4	5		5	4	11.11% 1	5	6		6	5	13.89% 1	6	7		7	6	16.67% 2	1	3		8	5	13.89% 2	2	4		9	4	11.11% 2	3	5		10	3	8.33% 2	4	6		11	2	5.56% 2	5	7		12	1	2.78% 2	6	8				3	1	4				3	2	5				3	3	6				3	4	7				3	5	8				3	6	9				4	1	5				4	2	6				4	3	7				4	4	8				4	5	9				4	6	10				5	1	6				5	2	7				5	3	8				5	4	9				5	5	10				5	6	11				6	1	7				6	2	8				6	3	9				6	4	10				6	5	11				6	6	12

Fake Nine Sided Die
Another way (which seems more intuitive/obvious to me) to produce uniformly random numbers from 1 to 9 via 2d6 is a table using the 2 rolls as independent indexes, e.g.:

1 2 3 4 5 6  - - - - - - 1: 1 1 2 2 2 2 2: 1 1 3 3 3 3 3: 4 4 5 5 5 5 4: 4 4 6 6 6 6 5: 7 7 8 8 8 8 6: 7 7 9 9 9 9

or (depending on your layout preference)

1 2 3 4 5 6  - - - - - - 1: 1 1 2 2 3 3 2: 1 1 2 2 3 3 3: 4 4 5 5 6 6 4: 4 4 5 5 6 6 5: 7 7 8 8 9 9 6: 7 7 8 8 9 9

although perhaps you could make it work this way, where the order of the combinations don't matter:

1 2 3 4 5 6  - - - - - - 1: 1 1 2 3 4 5 2: 1 1 2 3 4 5 3: 2 2 6 6 7 8 4: 3 3 6 6 7 8 5: 4 4 7 7 9 9 6: 5 5 8 8 9 9

Reroll Probability
The probability that you roll 4+ is the probability that you roll a 4, a 5, or a 6. Each of these events has probability 1/6; so, the probability of rolling 4+ is 3/6 = 1/2.

In the event that you get a 5+ when you are allowed on re-roll can be broken up in to two disjoint events: event "A", in which you roll a 5+ on the first try; and event "B", in which you roll a number 1-4 on your first attempt, and either a 5 or 6 on the second.

Then P(A) = 2/6 = 1/3, since this is the event that you roll either a 5 or a 6. For "B, we have P(B) = 4/6 ⋅ 2/6 = 2/9, since you must roll a 1, 2, 3, or 4 on the first attempt and either a 5 or 6 on the second. So, overall, the probability of getting 5+ when you allow one re-roll is P(A or B) = P(A) + P(B) = 1/3 + 2/9 = 5/9.

(Note that we have used here that "A" and "B" are disjoint possibilities.)

So, you are more likely to get a 5+ with a re-roll allowed than to get a 4+ with no re-roll.

Ref: How do i calculate Dice probability - Mathematics Stack Exchange - http://math.stackexchange.com/questions/478184/how-do-i-calculate-dice-probability

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1 of 6 (eg 6) roll 1 roll 2  roll 3  roll 4  roll 5  roll 6 16.67%	13.89%	11.57%	9.65%	8.04%	6.70% 16.67%	30.56%	42.13%	51.77%	59.81%	66.51%

2 of 6 (eg 5+) roll 1 roll 2  roll 3  roll 4  roll 5  roll 6 33.33%	22.22%	14.81%	9.88%	6.58%	4.39% 33.33%	55.56%	70.37%	80.25%	86.83%	91.22%

2 of 6 (eg 4+) roll 1 roll 2  roll 3  roll 4  roll 5  roll 6 50.00%	25.00%	12.50%	6.25%	3.13%	1.56% 50.00%	75.00%	87.50%	93.75%	96.88%	98.44%

Multiple Dice Probability
Roll 1 dice, we have 6 possibilities. Roll 2 dice, we have 36 possibilities. Roll 3 dice we have 216 possibilities... etc.

One thing that might help you understand probability with dice is say we roll 6 dice and want to know the probability of rolling a 6. Many people think...mistakenly...well, if it's a 1 in 6 chance to roll a 6 on each die, then there is a 100% probability of rolling a 6! Well, that's actually wrong. Since there are (6^6) 46,65646,656 possible outcomes, and (5^6) 15,625 do not have a 6, well then the probability is only (46,656 − 15,625) / 46,656 = 66.5%. Now I think you may have a better understanding.

Ref: How do i calculate Dice probability - Mathematics Stack Exchange - http://math.stackexchange.com/questions/478184/how-do-i-calculate-dice-probability

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My interpretation, as if they were 6 rerolls:
 * 1) (1/6) +
 * 2) (5/6) * (1/6) +
 * 3) (5/6) * (5/6) * (1/6) +
 * 4) (5/6) * (5/6) * (5/6) * (1/6) +
 * 5) (5/6) * (5/6) * (5/6) * (5/6) * (1/6) +
 * 6) (5/6) * (5/6) * (5/6) * (5/6) * (5/6) * (1/6) = 66.5%

Spin Down Dice
There are dice which roll the way you want (sometimes called barrel dice, Crystal Caste market them as 'crystal' dice). There are dice that count sequentially (e.g. Magic: The Gathering life counter dice).

https://www.reddit.com/r/boardgames/comments/47dznn/sequential_dicecounter/d0c9v1i/

https://i.imgur.com/10DNklW.png