Odds of dice..

With 10 dice and one flop of a cup...what are odds of getting same number with 8 of them???

my attempt
for standard 6 sided dice
10d6

the 2 patterns that would work for 8 of a kind would be
AAAAAAAA BB and AAAAAAAA BC
There are a total of 6^10 possible combinations = 60,466,176

AAAAAAAA = 10 choose 8 = 45 (10*9/2*1)
A could be a value of 1 thru 6
AAAAAAAA = 45*6 = 270

BB = 2 choose 2 = 1 and only 5 values B can take because it must be different from A
1*5=5

AAAAAAAA BB = 270*5= 1,350

BC = 5 values B can be and 4 values C can be.
BC = 5*4 = 20

AAAAAAAA BC = 270*20 = 5,400

6,750 = total number of ways to have 8 of a kind
total possible permutations = 60,466,176

probability = 6,750/60,466,176 = about 0.011163% or about 1 in 8,958

using different dice, maybe 10 sided ones, just use the values for the number of faces
should work every time as long as you consider every possible pattern that qualifies

Sally

computers easily can do the math for us too
a few tables of all the patterns for 10d6
sort by pattern

index	pattern	ways	probability	1 in
1	AABBCCDDEF	3402000	0.056262860	17.77
2	AABBCCDDEE	680400	0.011252572	88.87
3	AAABBCCDEF	9072000	0.150034294	6.67
4	AAABBCCDDE	9072000	0.150034294	6.67
5	AAABBBCDEF	1512000	0.025005716	39.99
6	AAABBBCCDE	9072000	0.150034294	6.67
7	AAABBBCCDD	2268000	0.037508573	26.66
8	AAABBBCCCD	1008000	0.016670477	59.99
9	AAAABBCDEF	2268000	0.037508573	26.66
10	AAAABBCCDE	6804000	0.112525720	8.89
11	AAAABBCCDD	1134000	0.018754287	53.32
12	AAAABBBCDE	3024000	0.050011431	20.00
13	AAAABBBCCD	4536000	0.075017147	13.33
14	AAAABBBCCC	252000	0.004167619	239.95
15	AAAABBBBCD	567000	0.009377143	106.64
16	AAAABBBBCC	189000	0.003125714	319.93
17	AAAAABCDEF	181440	0.003000686	333.26
18	AAAAABBCDE	1814400	0.030006859	33.33
19	AAAAABBCCD	1360800	0.022505144	44.43
20	AAAAABBBCD	907200	0.015003429	66.65
21	AAAAABBBCC	302400	0.005001143	199.95
22	AAAAABBBBC	151200	0.002500572	399.91
23	AAAAABBBBB	3780	0.000062514	15,996.34
24	AAAAAABCDE	151200	0.002500572	399.91
25	AAAAAABBCD	453600	0.007501715	133.30
26	AAAAAABBCC	75600	0.001250286	799.82
27	AAAAAABBBC	100800	0.001667048	599.86
28	AAAAAABBBB	6300	0.000104190	9,597.81
29	AAAAAAABCD	43200	0.000714449	1,399.68
30	AAAAAAABBC	43200	0.000714449	1,399.68
31	AAAAAAABBB	3600	0.000059537	16,796.16
32	AAAAAAAABC	5400	0.000089306	11,197.44
33	AAAAAAAABB	1350	0.000022327	44,789.76
34	AAAAAAAAAB	300	0.000004961	201,553.92
35	AAAAAAAAAA	6	0.000000099	10,077,696.00
	total	60,466,176

sort by largest # of ways

index	pattern	ways	probability	1 in
1	AAABBCCDEF	9072000	0.150034294	6.67
2	AAABBCCDDE	9072000	0.150034294	6.67
3	AAABBBCCDE	9072000	0.150034294	6.67
4	AAAABBCCDE	6804000	0.112525720	8.89
5	AAAABBBCCD	4536000	0.075017147	13.33
6	AABBCCDDEF	3402000	0.056262860	17.77
7	AAAABBBCDE	3024000	0.050011431	20.00
8	AAABBBCCDD	2268000	0.037508573	26.66
9	AAAABBCDEF	2268000	0.037508573	26.66
10	AAAAABBCDE	1814400	0.030006859	33.33
11	AAABBBCDEF	1512000	0.025005716	39.99
12	AAAAABBCCD	1360800	0.022505144	44.43
13	AAAABBCCDD	1134000	0.018754287	53.32
14	AAABBBCCCD	1008000	0.016670477	59.99
15	AAAAABBBCD	907200	0.015003429	66.65
16	AABBCCDDEE	680400	0.011252572	88.87
17	AAAABBBBCD	567000	0.009377143	106.64
18	AAAAAABBCD	453600	0.007501715	133.30
19	AAAAABBBCC	302400	0.005001143	199.95
20	AAAABBBCCC	252000	0.004167619	239.95
21	AAAABBBBCC	189000	0.003125714	319.93
22	AAAAABCDEF	181440	0.003000686	333.26
23	AAAAABBBBC	151200	0.002500572	399.91
24	AAAAAABCDE	151200	0.002500572	399.91
25	AAAAAABBBC	100800	0.001667048	599.86
26	AAAAAABBCC	75600	0.001250286	799.82
27	AAAAAAABCD	43200	0.000714449	1,399.68
28	AAAAAAABBC	43200	0.000714449	1,399.68
29	AAAAAABBBB	6300	0.000104190	9,597.81
30	AAAAAAAABC	5400	0.000089306	11,197.44
31	AAAAABBBBB	3780	0.000062514	15,996.34
32	AAAAAAABBB	3600	0.000059537	16,796.16
33	AAAAAAAABB	1350	0.000022327	44,789.76
34	AAAAAAAAAB	300	0.000004961	201,553.92
35	AAAAAAAAAA	6	0.000000099	10,077,696.00
	total	60,466,176

Quote: mustangsally

my attempt
for standard 6 sided dice
10d6

the 2 patterns that would work for 8 of a kind would be
AAAAAAAA BB and AAAAAAAA BC
There are a total of 6^10 possible permutations = 60,466,176

Are we sure about this he said 8 of the same so did he want exactly 8 or did he want 8 or more. Yours is right for exactly 8.

Quote: Twirdman

Are we sure about this he said 8 of the same so did he want exactly 8 or did he want 8 or more. Yours is right for exactly 8.

Not really sure but to add in the ways to get 9 of a kind and 10 of a kind is real easy.

10oak = 10 choose 10 = 1 * 6 = 6 ways
9oak = 10 choose 9 = 10 *6 (A could be any of 6 values) * 5 for B = 300 ways

only adding another 306 ways for 8 of a kind or higher to the 6,750 for just 8 of a kind

wondering what game this might be for.
other than just a math question

Sally

Quote: mustangsally

Not really sure but to add in the ways to get 9 of a kind and 10 of a kind is real easy.

10oak = 10 choose 10 = 1 * 6 = 6 ways
9oak = 10 choose 9 = 10 *6 (A could be any of 6 values) * 5 for B = 300 ways

only adding another 306 ways for 8 of a kind or higher to the 6,750 for just 8 of a kind

wondering what game this might be for.
other than just a math question

Sally

Oh yeah I know just wanted to make sure the person knew there were 2 different answers depending on exactly what his question was and yeah cannot think exactly what game this would be. I mean could technically be for just a pen and paper RPG I have played games where what matters is the number of matches you get and you roll a lot of dice. Though don't recall ever having to roll 10 dice at once for a check like that.

That is also why I pointed out the more than 8 thing in those games you get a success if you get exactly 8 or if you get more than 8.

i got the same answer.

6*[10C2*(1/6)^8*(5/6)^2)]=0.00011163266

Thank you so much...it is 10-six sided die....need 8(or more-8 would make you the big winner) of the same number to come up....one flop....you say 1:8,958, the chart seems to say 1:11,197....????? By the way...the game is a "flop of the day" at a local Pub...jackpot is up to $7,000(No one is supposed to know what is in the jackpot when you agree to try-for some reason the owner told people), it costs $1 to try.....