Dice Odds rolling 8 of a kind on one shake with 10 dice

Hi Wizard, I am new to this site I like craps and am studying the game. I am a pilot and not a math expert but enjoy it. But my question now is, and this part is important to me, I would like to know the odds for my 10 dice question but more importantly want to understand how to put into formula and calculate. I have searched the internet and found different answers, from 1 in 37000 to 1 in 8500. I think I understand how to calculate the odds of 8 dice 8 of a kind equals 1 in 1,679,916, 6 to the 8th power, but not sure if that is for a specific number ie 8, 6's or any 8 of a kind. Now throw in 2 extra dice I am lost. I have a calculator that has the nCr function which is for combinations I think. Or maybe I need to use the nPr function. One formula I found is 6*combin(10,8)*(1/6) to the 8th*(5/6) to the 2nd = 1/8957.952 Not sure what the numbers really represent or how to plug into calculator which is what I really want to know. So then I could figure out other odds. Thanks for any help. Great site!
Bob

Oh shoot I didn't specify any 8 of a kind vs a specific 8 of a kind, would love to see how both are calculated.

First of all, I am assuming you are using six-sided dice.

If you want exactly 8 of the same number (as opposed to 8 or more), but it can be any number:
(a) There are 6 possibilities for the number (e.g. eight 1s, or eight 6s)
(a) There are 10C2 = 45 ways of selecting the two dice that are not in the 8 of a kind;
(b) There are 5 different numbers that the "first" of these two dice can be;
(c) There are 5 different numbers that the "second" of these two dice can be;

There are a total of 6 * 45 * 5 * 5 = 6750 different rolls that can be eight of a kind, out of the 6¹⁰ = 60,466,176 different possible rolls; the probability is 1 in (60,466,176 / 6750) = 1 in 8958. This is one of the formulas you found.

If you want a specific 8 of a kind (e.g. it has to be eight 6s), then ignore (a) in the above, so there are only 1125 possible rolls, and the probability is 1 in (60,466,176 / 1225) = 1 in 53,748.

I am not sure where you got "37,000 to 1."

Thanks a lot ThatDonGuy! I will study this! Might have a few more questions. But really appreciate it! The 37000!to 1 was one of a few different answers I found when searched internet. Oh and yes standard 6 sided dice. How would it look if I wanted 8 or more? i really became curious when I was at a place in WI that had a pot up to $10k if matched 8 out of 10 with 1 shake. Thanks again.
PS is it correct to match 8 of a kind on one shake with 8 dice is 6 to the 8th power? If correct is that for a specific number or any 8 of a kind?
Bob

Quote: Rltkktlr

How would it look if I wanted 8 or more? i really became curious when I was at a place in WI that had a pot up to $10k if matched 8 out of 10 with 1 shake. Thanks again.

Is it correct to match 8 of a kind on one shake with 8 dice is 6 to the 8th power? If correct is that for a specific number or any 8 of a kind?

To calculate 8 or more, calculate 8, 9, and 10 separately, and then add them up.
9 of a kind is 6 (there are 6 different numbers that the 9 of a kind can be) * 10C1 (since you need to choose the one die that isn't in the 9 of a kind) * 5 (since there are 5 possible numbers it can be) = 450.
(You can also use 6 * 10C9 (to choose the 9 dice in the 9 of a kind rather than the 1 die that's not in it) * 5; this is also 450. Note that nCr = nC(n-r), since there are just as many ways to choose r items out of n as there are "not to choose" (n-r) items out of n - for every set of r items you choose, there is a corresponding set of (n-r) items that you did not choose.)

8 of a kind out of 8 with one shake is 1 in 6⁸ for a specific number, or 6 times that for any 8 of a kind. There are 6⁸ ways to roll 8 dice; one of them is, say, eight 6s, and six of them (1s, 2s, 3s, 4s, 5s, 6s) are 8 of a kind.

This method can be done in a spreadsheet or I do know there are calculators that also can do this.
I think I have used this one before
http://blog.plover.com/math/yahtzee.html
He also does a nice job of explaining how to count the number of different ways of rolling a certain
pattern of numbers with a certain number of dice

The distribution is this for 10d6 patterns

pattern	ways	probability
AABBCCDDEF	3,402,000	0.05626286
AABBCCDDEE	680,400	0.011252572
AAABBCCDEF	9,072,000	0.150034294
AAABBCCDDE	9,072,000	0.150034294
AAABBBCDEF	1,512,000	0.025005716
AAABBBCCDE	9,072,000	0.150034294
AAABBBCCDD	2,268,000	0.037508573
AAABBBCCCD	1,008,000	0.016670477
AAAABBCDEF	2,268,000	0.037508573
AAAABBCCDE	6,804,000	0.11252572
AAAABBCCDD	1,134,000	0.018754287
AAAABBBCDE	3,024,000	0.050011431
AAAABBBCCD	4,536,000	0.075017147
AAAABBBCCC	252,000	0.004167619
AAAABBBBCD	567,000	0.009377143
AAAABBBBCC	189,000	0.003125714
AAAAABCDEF	181,440	0.003000686
AAAAABBCDE	1,814,400	0.030006859
AAAAABBCCD	1,360,800	0.022505144
AAAAABBBCD	907,200	0.015003429
AAAAABBBCC	302,400	0.005001143
AAAAABBBBC	151,200	0.002500572
AAAAABBBBB	3,780	6.25143E-05
AAAAAABCDE	151,200	0.002500572
AAAAAABBCD	453,600	0.007501715
AAAAAABBCC	75,600	0.001250286
AAAAAABBBC	100,800	0.001667048
AAAAAABBBB	6,300	0.00010419
AAAAAAABCD	43,200	0.000714449
AAAAAAABBC	43,200	0.000714449
AAAAAAABBB	3,600	5.95374E-05
AAAAAAAABC	5,400	8.93061E-05
AAAAAAAABB	1,350	2.23265E-05
AAAAAAAAAB	300	4.96145E-06
AAAAAAAAAA	6	9.9229E-08
total	60,466,176	1

Don made a SallyOh error for the 9oak. should = 300 (6*10*5)

super high house edge for the game you saw!
Sally

a shake
I love chocolate shakes and so do my hips and thighs

here is a shake game I found too

at least it is not this game

Thanks Don and Sally, great song by the way! Yes thanks for showing the 300 as the correct answer. Would have really confused me! But still lacking correlation! Not sure how to plug 300 into original formula. Or would it be 900 (300 for 8,9,10) anyway not sure what to do with the 300(900). Need to know next step. Also to help me with further understanding, back to matching any 8 of a kind but this time shaking 12 dice? I tried what I thought was maybe correct and came up with about 1 in 36. Seems way off. Thanks again.
Bob

Thanks Don and Sally, great song by the way! Yes thanks for showing the 300 as the correct answer. Would have really confused me! But still lacking correlation! Not sure how to plug 300 into original formula. Or would it be 900 (300 for 8,9,10) anyway not sure what to do with the 300(900). Need to know next step. Also to help me with further understanding, back to matching any 8 of a kind but this time shaking 12 dice? I tried what I thought was maybe correct and came up with about 1 in 36. Seems way off. Thanks again.
Bob
Ps sorry again just trying to learn what I should have learned in collage!

Quote: Rltkktlr

Not sure how to plug 300 into original formula. Or would it be 900 (300 for 8,9,10) anyway not sure what to do with the 300(900). Need to know next step.

you are so close.
took me a few years to finally get this type of math
Don might have been doing these dice problems as a two year old. Some are so lucky!
The link I also provided I thought really explains well how to solve these type of pattern problems.

here
what you need to understand is the pattern
your example:
AAAA AAAABC: 5400 ways
Is one way to get 8oak with 2 singletons for the other two dice
AAAA AAAABB:1350
Is another way to get 8oak with a pair for the other two dice
AAAA AAAA AB: 300
the only way to get 9oak
A can bet any of 6 values
B can be any of 5 values
because we Have 10 and Want 9oak 10C9 = 10
6*5*10=300
AAAA AAAA AA: 6

so for any 8,9 or 10oak we sum each possible pattern
that should be clear

one more example
AAAA AAAABC: 5400 ways
we Have 10 and Want 8oak = 10C8 = 10! / 8! (10-2)! = 10*9/2*1 = 45
A could be any 6 values
B must be 5 values
C must be 4 values
45*6*5*4

Quote: Rltkktlr

Also to help me with further understanding, back to matching any 8 of a kind but this time shaking 12 dice? I tried what I thought was maybe correct and came up with about 1 in 36. Seems way off. Thanks again.
Bob
Ps sorry again just trying to learn what I should have learned in collage!

This is really high school math.
maybe with the kids today, grade school even.

here is the table for the 12 dice

pattern	ways	probability
AABBCCDDEEFF	7,484,400	0.003438286
AAABBCCDDEEF	149,688,000	0.068765718
AAABBBCCDDEF	299,376,000	0.137531436
AAABBBCCDDEE	99,792,000	0.045843812
AAABBBCCCDEF	44,352,000	0.020375028
AAABBBCCCDDE	133,056,000	0.061125083
AAABBBCCCDDD	5,544,000	0.002546878
AAAABBCCDDEF	149,688,000	0.068765718
AAAABBCCDDEE	37,422,000	0.017191429
AAAABBBCCDEF	199,584,000	0.091687624
AAAABBBCCDDE	299,376,000	0.137531436
AAAABBBCCCDE	99,792,000	0.045843812
AAAABBBCCCDD	49,896,000	0.022921906
AAAABBBBCDEF	12,474,000	0.005730476
AAAABBBBCCDE	74,844,000	0.034382859
AAAABBBBCCDD	18,711,000	0.008595715
AAAABBBBCCCD	24,948,000	0.011460953
AAAABBBBCCCC	693,000	0.00031836
AAAAABBCCDEF	59,875,200	0.027506287
AAAAABBCCDDE	59,875,200	0.027506287
AAAAABBBCDEF	19,958,400	0.009168762
AAAAABBBCCDE	119,750,400	0.055012574
AAAAABBBCCDD	29,937,600	0.013753144
AAAAABBBCCCD	19,958,400	0.009168762
AAAAABBBBCDE	19,958,400	0.009168762
AAAAABBBBCCD	29,937,600	0.013753144
AAAAABBBBCCC	3,326,400	0.001528127
AAAAABBBBBCD	2,993,760	0.001375314
AAAAABBBBBCC	997,920	0.000458438
AAAAAABBCDEF	9,979,200	0.004584381
AAAAAABBCCDE	29,937,600	0.013753144
AAAAAABBCCDD	4,989,600	0.002292191
AAAAAABBBCDE	13,305,600	0.006112508
AAAAAABBBCCD	19,958,400	0.009168762
AAAAAABBBCCC	1,108,800	0.000509376
AAAAAABBBBCD	4,989,600	0.002292191
AAAAAABBBBCC	1,663,200	0.000764064
AAAAAABBBBBC	665,280	0.000305625
AAAAAABBBBBB	13,860	6.3672E-06
AAAAAAABCDEF	570,240	0.000261965
AAAAAAABBCDE	5,702,400	0.002619646
AAAAAAABBCCD	4,276,800	0.001964735
AAAAAAABBBCD	2,851,200	0.001309823
AAAAAAABBBCC	950,400	0.000436608
AAAAAAABBBBC	475,200	0.000218304
AAAAAAABBBBB	23,760	1.09152E-05
AAAA AAAABCDE	356,400	0.000163728
AAAA AAAABBCD	1,069,200	0.000491184
AAAA AAAABBCC	178,200	8.18639E-05
AAAA AAAABBBC	237,600	0.000109152
AAAA AAAABBBB	14,850	6.822E-06
AAAA AAAA ABCD	79,200	3.6384E-05
AAAA AAAA ABBC	79,200	3.6384E-05
AAAA AAAA ABBB	6,600	3.032E-06
AAAA AAAA AABC	7,920	3.6384E-06
AAAA AAAA AABB	1,980	9.09599E-07
AAAA AAAA AAAB	360	1.65382E-07
AAAA AAAA AAAA	6	2.75636E-09
totals	2,176,782,336	1

for 8oak there looks to be 5 patterns (that to me is the hard part just accounting for each pattern that could be thrown - that is why the letters are used and why I use the computer)
look at each pattern and see if you can do the math for each pattern

AAAA AAAA BCDE
start with you Have 12 and Want 8 = 12C8
Now you have 4 and want ?
go until you have none left
the value each letter could be how many and multiply them all for each
then sum them up from all the number of patterns

if you really want to learn this, try using 3 or 4 dice to start
that way you can see quickly how you are doing and it will build confidence.

have fun!
Sally

Thanks Sally. Yes probably high school level! My problem is my girlfriend was in my stats class in collage so failed to pay attention! Yes I know waste of my tuition! Ok I will study up. See if I can make heads or tails of it! Shoot missed your link I will look at it. Thanks
Bob.
Ps could be more questions but not until I study!

Cool looking shaking table game! What's it's called?