Probability Thread

[Max_Hult]

Whats the odds of a edge pairing skip using yau? (4x4) i mean like finishing centers and getting straight to 3x3 stage
It happened me today and i feel like it must very insanely rare

 

[xyzzy]

Whats the odds of a edge pairing skip using yau? (4x4) i mean like finishing centers and getting straight to 3x3 stage
It happened me today and i feel like it must very insanely rare

Since you're using Yau, that means you start off with 3 cross edge pairs solved before you finish solving the centres, which leaves you with 9 edge pairs to form, using 18 (individual) edge pieces.

Pick any one of those 18 edge pieces; there are 17 choices for what its partner could be, and only 1 of them leads to an edge pair. (1/17 chance)
Pick any one of the remaining 16 edge pieces; there are 15 choices for what its partner could be, and only 1 of them leads to an edge pair. (1/15 chance)
etc. etc.

The probability of all 18 edges being paired up is thus 1/(17×15×13×…5×3×1) = 1/34459425.

(see also)

(complication/caveat: This requires making an assumption that you always solve the same 3 cross edges after F2C and you never go out of your way to preserve edge pairs. If not, the probabilities involved will be distorted by preferentially choosing already-formed edge pairs to use in the cross / preserving other already-formed pairs.)

 

[Bruce MacKenzie]

I sat down and wrote some code to count the number of ways to have two solved 2x2x1 blocks on adjacent faces. On a face there are four 2x2x1 blocks which may be turned four ways to give 16 patterns. For two adjacent faces the blocks may be combined in 144 ways which are disjoint. That is the blocks do not intersect. Applying the 48 rotational and mirror symmetries gives 1728 ways to apply these patterns to the cube as a whole. Many of the original patterns have symmetry or are symmetry conjugates of one another so one does not get the full 48 times the number of patterns.

So, 1728 / 95,800,320 = 1 in 55,440 chances of getting two solved 2x2x1 blocks on adj. faces of the cube in a random scramble.

Still playing around with this, I partitioned the 1728 patterns into 78 equivalence classes by conjugation with the 24 rotation symmetries. I then picked an element from each class having the two solved blocks on the Front and Right faces to represent the class. This element actually defines a subgroup of the cube group having some 4.5 x 10^11 elements depending on the arrangement of the other 14 cubies. I picked a random element from each subgroup and solved it, giving a turn sequence to illustrate the class. These results are listed in the following table.

​	​	Template(Singmaster Facelet Permutation)​	​
Class​	Size​	UF UR UB UL DF DR DB DL FR FL BR BL UFR URB UBL ULF DRF DFL DLB DBR​	Instance​
1​	12​	LF •• •• •• •• BR •• •• •• FD UR •• ••• ••• ••• LDF ••• ••• ••• BUR​	F R F D L' D2 B' L D' F L' B U F R F D R'​
2​	12​	•• UR •• •• DF •• •• •• •• FL BR •• ••• URB ••• ••• ••• DFL ••• •••​	R' U2 R U D B U' D' F' B' L' U' F U2 B R U'​
3​	12​	UF •• •• •• •• DR •• •• •• FL BR •• ••• ••• ••• ULF ••• ••• ••• DBR​	R' F2 B' L F' B D L D F2 R2 U B' U B R' F​
4​	12​	•• BR •• •• LF •• •• •• •• FU DR •• ••• BRD ••• ••• ••• LFU ••• •••​	F' B' U F R F D' L' F' U2 D F' B L' F R' F' L'​
5​	24​	LF DR •• •• •• •• •• •• BR FD •• •• DBR ••• ••• LDF ••• ••• ••• •••​	F' L' B L B' L U2 F' D' L' F' R' D B R' B D2​
6​	24​	RF •• •• •• •• DR •• •• •• FU BR •• ••• ••• ••• RUF ••• ••• ••• DBR​	L' U R2 B' R' L U' B' L U2 R' U L' B U L' U' F' B'​
7​	24​	•• FR •• •• DF •• •• •• •• FL UR •• ••• FRU ••• ••• ••• DFL ••• •••​	R' B R D' R2 U' F U R' D F' L' D' L' F D F D'​
8​	24​	LF DR •• •• •• •• •• •• FU •• FR •• LFU DRF ••• ••• ••• ••• ••• •••​	L U B U' R L' D' B' R D' B R D R' F B2 L' B​
9​	24​	DF •• •• •• •• DR •• •• FL •• BR •• DFL ••• ••• ••• ••• ••• ••• DBR​	U F' U2 L U' D L' U' L' U' D' L' U F' B' L B​
10​	24​	DF UR •• •• •• •• •• •• FL •• BR •• DFL URB ••• ••• ••• ••• ••• •••​	U' R' U R U L' F B D F' D' B D L U F' U F' L'​
11​	24​	LF •• •• •• •• FR •• •• FU •• DR •• LFU ••• ••• ••• ••• ••• ••• FDR​	B' L D' B' D L U R' F L U' F' R' D R U'​
12​	24​	•• BR •• •• UF •• •• •• •• FR DR •• ••• BRD ••• ••• ••• UFR ••• •••​	R' U' D R' U' F R' B' L F D2 B L F U D R L​
13​	24​	•• DR •• •• RF •• •• •• BR FD •• •• DBR ••• ••• ••• ••• RFD ••• •••​	U' R L B2 R' U' L U' R B R' L' F R B' U' R​
14​	24​	LF UR •• •• •• •• •• •• FR FD •• •• UFR ••• ••• LDF ••• ••• ••• •••​	L U L' U' F B' D R B R' B' R L B L' F' R' U' F R'​
15​	24​	DF •• •• •• •• BR •• •• FL •• UR •• DFL ••• ••• ••• ••• ••• ••• BUR​	F R F L' F L' B' U B' D L' D R' F R D2 L' U F' B'​
16​	24​	UF FR •• •• •• •• •• •• •• FL UR •• ••• FRU ••• ULF ••• ••• ••• •••​	D' F L' F' D B D' F L2 F' R F L' D2 L D' F'​
17​	24​	•• BR •• •• RF •• •• •• •• FD DR •• ••• BRD ••• ••• ••• RFD ••• •••​	F U' F' L F' U' L' U F L U' F R B' U B R B R​
18​	24​	RF UR •• •• •• •• •• •• FD •• BR •• RFD URB ••• ••• ••• ••• ••• •••​	U F' D' R D' R' D L' U L B' L F R' F D F2 D​
19​	24​	LF •• •• •• •• FR •• •• •• FD DR •• ••• ••• ••• LDF ••• ••• ••• FDR​	R2 B D R U L' D' L U' B R' B' D' F R' D B' R​
20​	24​	RF UR •• •• •• •• •• •• •• FU BR •• ••• URB ••• RUF ••• ••• ••• •••​	L' D L F' B' D' B' D B R' L' D R L D L' D L​
21​	24​	•• DR •• •• UF •• •• •• BR FR •• •• DBR ••• ••• ••• ••• UFR ••• •••​	U' R' D R' D B D' R L F' L F' D F R' B' L' D B​
22​	24​	DF •• •• •• •• BR •• •• DR FR •• •• ••• ••• ••• DRF BRD ••• ••• •••​	F' R F D L2 U D' L D F' L' D' F' U' R B2 D2 R'​
23​	24​	DF DR •• •• •• •• •• •• FL •• FR •• DFL DRF ••• ••• ••• ••• ••• •••​	R' F R' D R' D' B' R' B R2 F B R D' R' B' D L'​
24​	24​	RF •• •• •• •• DR •• •• FD •• BR •• RFD ••• ••• ••• ••• ••• ••• DBR​	R' U R L' B' L B' U2 R' L U L' F' L' F R U' L' F'​
25​	24​	•• BR •• •• UF •• •• •• FL •• DR •• ••• BRD ••• ••• ULF ••• ••• •••​	U' R F L F' B R D' R D' F' U' L U R' B R' B R2 F'​
26​	24​	•• BR •• •• LF •• •• •• FD •• DR •• ••• BRD ••• ••• LDF ••• ••• •••​	F' U B L F B' U L2 F' R' D F' R' L' D' B R L' D'​
27​	24​	•• UR •• •• DF •• •• •• FR FL •• •• UFR ••• ••• ••• ••• DFL ••• •••​	R' U2 L D' R D L B' R' F' B U' L' F U' B' U R' L'​
28​	24​	DF FR •• •• •• •• •• •• FL •• UR •• DFL FRU ••• ••• ••• ••• ••• •••​	B R L D B U D' L' U B' U2 B' R' D' R D F2​
29​	24​	•• BR •• •• UF •• •• •• UR FR •• •• BUR ••• ••• ••• ••• UFR ••• •••​	D2 F B' L U L U' B L' U L' U' L F D' R' D​
30​	24​	LF UR •• •• •• •• •• •• FU •• BR •• LFU URB ••• ••• ••• ••• ••• •••​	F' R2 U2 R B U' D L U' L' B' D' R D B' D' F2​
31​	24​	DF UR •• •• •• •• •• •• •• FR BR •• ••• URB ••• DRF ••• ••• ••• •••​	F' R U F' R U' L' U' F B U F' R L' D B' L D B D' R​
32​	24​	UF BR •• •• •• •• •• •• FR •• DR •• UFR BRD ••• ••• ••• ••• ••• •••​	F R U R B' L F' D' F D' R' F' L2 U L F B' R U' R​
33​	24​	LF FR •• •• •• •• •• •• FU •• UR •• LFU FRU ••• ••• ••• ••• ••• •••​	D B' D B R U F D R L B L' F' U' B' R' F L'​
34​	24​	LF •• •• •• •• BR •• •• DR FD •• •• ••• ••• ••• LDF BRD ••• ••• •••​	F R' L F' B' R U R' B R' B R' B U' L' F' L F B'​
35​	24​	UF FR •• •• •• •• •• •• DR FL •• •• FDR ••• ••• ULF ••• ••• ••• •••​	U' R' B' U' B R B U B R D B2 R' B R D' B​
36​	24​	RF BR •• •• •• •• •• •• FD •• DR •• RFD BRD ••• ••• ••• ••• ••• •••​	F R2 U2 D' F' R' F' U L' U' L' B U' F L2 B L' F'​
37​	24​	LF DR •• •• •• •• •• •• •• FD FR •• ••• DRF ••• LDF ••• ••• ••• •••​	L U B R U' B D B' U R2 L' U F2 U L F R'​
38​	24​	DF •• •• •• •• UR •• •• BR FR •• •• ••• ••• ••• DRF URB ••• ••• •••​	F' D F' R' L' D L B' R' B D' B2 U' D B' U​
39​	24​	UF UR •• •• •• •• •• •• FR •• BR •• UFR URB ••• ••• ••• ••• ••• •••​	L D' L B' D' R L D R' B' L' U B' U' D L' B L' D​
40​	24​	DF •• •• •• •• DR •• •• •• FR BR •• ••• ••• ••• DRF ••• ••• ••• DBR​	L U2 R B' R' L' F L' F' U R B' R' F' L' F' U R B' R'​
41​	24​	LF •• •• •• •• FR •• •• UR FD •• •• ••• ••• ••• LDF FRU ••• ••• •••​	F R F D' R L' U' F U2 B' L F B U R D L U B​
42​	24​	•• BR •• •• DF •• •• •• FR •• DR •• ••• BRD ••• ••• DRF ••• ••• •••​	L' U D' L U' D F' R' D' R F' L U' R' L' F D F​
43​	24​	LF FR •• •• •• •• •• •• •• FD UR •• ••• FRU ••• LDF ••• ••• ••• •••​	R' U B' U R' U' R F' U' L U F B' R D' R D F D'​
44​	24​	DF •• •• •• •• BR •• •• •• FR UR •• ••• ••• ••• DRF ••• ••• ••• BUR​	U' R B' D B' D' R' B L' B' U' F2 B D' B' R B R'​
45​	24​	•• BR •• •• RF •• •• •• UR FD •• •• BUR ••• ••• ••• ••• RFD ••• •••​	R' L D' B' U D F' L' F D F' D' R F L F U' F' L'​
46​	24​	LF •• •• •• •• DR •• •• FU •• BR •• LFU ••• ••• ••• ••• ••• ••• DBR​	R D' L U B' D B' U' L B2 L2 F L' U' F' R' F​
47​	24​	UF DR •• •• •• •• •• •• BR FL •• •• DBR ••• ••• ULF ••• ••• ••• •••​	L' F' L U' L' F U L F D F' D' R L' U' R L B' D'​
48​	24​	RF DR •• •• •• •• •• •• BR FU •• •• DBR ••• ••• RUF ••• ••• ••• •••​	F' R L2 F' R U B D2 B2 U' R' F B' R B D'​
49​	24​	UF UR •• •• •• •• •• •• FR FL •• •• UFR ••• ••• ULF ••• ••• ••• •••​	R' U2 D' B D B' L B R' L' F' L' U R' F R' D B' D​
50​	24​	UF UR •• •• •• •• •• •• •• FL BR •• ••• URB ••• ULF ••• ••• ••• •••​	F' R U L U' R' L' F2 D' R' B R D' F' D F L' F' D​
51​	24​	DF DR •• •• •• •• •• •• BR FR •• •• DBR ••• ••• DRF ••• ••• ••• •••​	F R U B R L' F' L U' R D L F' R' F U L' F L' D' L​
52​	24​	•• FR •• •• DF •• •• •• DR FL •• •• FDR ••• ••• ••• ••• DFL ••• •••​	U' R F D' B' D F' B' R' U' B' U' B U' F' B D L D' F R​
53​	24​	LF •• •• •• •• DR •• •• •• FD BR •• ••• ••• ••• LDF ••• ••• ••• DBR​	R F' L U L F R B U' R' U R' F' U' L F' L2 F'​
54​	24​	•• BR •• •• DF •• •• •• •• FL DR •• ••• BRD ••• ••• ••• DFL ••• •••​	L U2 F' U L F R B' L B L' U B' R B R L' F' L' F​
55​	24​	•• DR •• •• DF •• •• •• •• FL FR •• ••• DRF ••• ••• ••• DFL ••• •••​	F B2 D R' U L D' B' L D' R' F2 U' D' F D' F​
56​	24​	LF BR •• •• •• •• •• •• •• FD DR •• ••• BRD ••• LDF ••• ••• ••• •••​	F R2 U' B' L' B D' L F' L F L D B L U D' R B' L'​
57​	24​	LF UR •• •• •• •• •• •• •• FD BR •• ••• URB ••• LDF ••• ••• ••• •••​	L' D' L' F' B U R' D L D' R' L U' D B D L D L'​
58​	24​	•• DR •• •• DF •• •• •• BR FL •• •• DBR ••• ••• ••• ••• DFL ••• •••​	F' R U B R' U' L' U F' L' U' F2 R F' B' L F R​
59​	24​	LF •• •• •• •• DR •• •• FR FD •• •• ••• ••• ••• LDF DRF ••• ••• •••​	U' R' U R U F' L' B L' F R B' R F R' B' D B D' R'​
60​	24​	DF BR •• •• •• •• •• •• UR FR •• •• BUR ••• ••• DRF ••• ••• ••• •••​	R' D2 R' B2 R B' R' U R' D' L' D F R U R L'​
61​	24​	DF •• •• •• •• FR •• •• FL •• DR •• DFL ••• ••• ••• ••• ••• ••• FDR​	U2 R' F R L U L' F' D' B D F R2 D' F L' D' R​
62​	24​	LF FR •• •• •• •• •• •• DR FD •• •• FDR ••• ••• LDF ••• ••• ••• •••​	U' R2 F D' R' F B2 L F' L D L B D' L' D R' B' R​
63​	24​	•• BR •• •• RF •• •• •• FU •• DR •• ••• BRD ••• ••• RUF ••• ••• •••​	L F2 R' D' F R' D' B' L' B' R2 L' F' U R' F'​
64​	24​	DF BR •• •• •• •• •• •• •• FR DR •• ••• BRD ••• DRF ••• ••• ••• •••​	F2 U R D' B U D B' U D' B U' B D B U' R2​
65​	24​	LF BR •• •• •• •• •• •• FU •• DR •• LFU BRD ••• ••• ••• ••• ••• •••​	R2 D L' D L B L D B' L2 D L D2 L' D' R F​
66​	24​	UF BR •• •• •• •• •• •• UR FL •• •• BUR ••• ••• ULF ••• ••• ••• •••​	F' B' L' B R' L B2 U D B D' B U' R F R' D'​
67​	24​	RF BR •• •• •• •• •• •• UR FU •• •• BUR ••• ••• RUF ••• ••• ••• •••​	R2 U R' D L' U' B' D R U' D F' U' B U B L U B U​
68​	24​	•• BR •• •• DF •• •• •• UR FL •• •• BUR ••• ••• ••• ••• DFL ••• •••​	R2 U' B U F' L F U B' U B' R' U' R B' R B​
69​	24​	RF BR •• •• •• •• •• •• •• FU DR •• ••• BRD ••• RUF ••• ••• ••• •••​	R2 U B D' R B' D' R' U' F B' L F' B' L2 D' R F'​
70​	24​	LF BR •• •• •• •• •• •• UR FD •• •• BUR ••• ••• LDF ••• ••• ••• •••​	R2 U B' U' R F B' D F L' U' D' L U F' D B D' B​
71​	24​	UF BR •• •• •• •• •• •• •• FL DR •• ••• BRD ••• ULF ••• ••• ••• •••​	L' U' L' U F' R' D L' D F U D' L B D2 L B' U'​
72​	24​	DF BR •• •• •• •• •• •• FL •• DR •• DFL BRD ••• ••• ••• ••• ••• •••​	F' R L U' B' L' D' R' B R2 B R' D B R' L U R' F'​
73​	24​	UF •• •• •• •• DR •• •• FR FL •• •• ••• ••• ••• ULF DRF ••• ••• •••​	D R' F B' L D B L2 B2 D' B D' L' F' R B'​
74​	24​	UF DR •• •• •• •• •• •• •• FL FR •• ••• DRF ••• ULF ••• ••• ••• •••​	R' U R U' R B' R' B R B' R' B R D B' R D​

 

[OldSwiss]

Only half cubing related:

I was searching the internet for a software development problem and on a microsoft Q&A I find a post of @David Zemdegs, having almost the same problem as I.
It could be the a different person with the same name but I guess not with this avatar



How crazy. There are 8 billion prople on this planet and you accidently find a post of someone you know.
What's the probabilty of this?

 

[David Zemdegs]

Only half cubing related:

I was searching the internet for a software development problem and on a microsoft Q&A I find a post of @David Zemdegs, having almost the same problem as I.
It could be the a different person with the same name but I guess not with this avatar

View attachment 24792

How crazy. There are 8 billion prople on this planet and you accidently find a post of someone you know.
What's the probabilty of this?

And I found the answer for you.

 

[OldSwiss]

And I found the answer for you.

Unfortunatley not. It was a similiar Problem but not exactly the same.
I've got no results from the three view and you got some but not all.

 

[JDAW6]

Probability for a scramble with all edges solved on 3x3

 

[Bruce MacKenzie]

Probability for a scramble with all edges solved on 3x3

12! • 2^11 = 980,995,276,800

 

[Mastermind2368]

Only half cubing related:

I was searching the internet for a software development problem and on a microsoft Q&A I find a post of @David Zemdegs, having almost the same problem as I.
It could be the a different person with the same name but I guess not with this avatar

View attachment 24792

How crazy. There are 8 billion prople on this planet and you accidently find a post of someone you know.
What's the probabilty of this?

While it is always neat to find people you know in unexpected places, I don't think it's honestly as rare as it may initially seem. These are rough estimates, but I'll try to work this out. First, lets assume some facts.

Spoiler: Facts assumed

• ~5 billion people who have access to the internet (demandsage)
  • Lets use 4 billion though, to account for people who lurkers and people who you would never interact with irl or online (people who are only on weird parts of the internet for example, still this is probably a high estimate)
• You've surfed the internet for ~5 years (Adjust as needed)
• You surf the internet 300 days out of the year (This is completely arbitrary and is based upon my own usage, change the number as needed)
• Each day on the internet, you encounter 100 "random" people (Suggested posts on social media, comments, replies, forum posts for random things, ect... For example, just one mouse wheel scroll on YouTube comments accounts for 15 people)
• There are 200 people who are closer to you than David Zemdegs, or are as close to you as David Zemdegs is (People you know at school, extended family, other cubers you know on here, ect...)

Spoiler: Calculations

• Days on the internet = 5*300=1500
• Total internet encounters = Internet days * 100 = 1500*100 = 1.5*10^5
• Percentage of internet users you know = People you know / internet population = (200/(4.0*10^9))*100=0.000005%
• Chance you've seen someone you know online = Percentage of users you know * total internet encounters = 0.75% (Something might be off on this step, someone should double check)
• Here's a desmos link. You can change the variables to better suit your own personal estimates.

 

[joshisbored]

i got a double x-cross with a ll skip ( misscramble :[[[[ )

 

[Jorian Meeuse]

i got a double x-cross with a ll skip ( misscramble :[[[[ )

I got the official pyraminx WR with a time of 0.19
Misscramble, or should I say, noscramble

 

[joshisbored]

lol
i was a scrambler at my last comp, and the runners sometimes mess up the solved and scrambled cubes and you get an unscrambled cube

 

[Thom S.]

How crazy. There are 8 billion prople on this planet and you accidently find a post of someone you know.

I don't think it's honestly as rare as it may initially seem.

I think if one has been on the internet for long enough, such coincidents will start to pile up.

 

[OldSwiss]

While it is always neat to find people you know in unexpected places, I don't think it's honestly as rare as it may initially seem. These are rough estimates, but I'll try to work this out. First, lets assume some facts.

Interesting calculation. Thanks for that.

On the other side there are only few people using their real name in the internet or at least the same alias across different websites.
I've probably met 100 more people I already know from somewhere else but don't recognize them

 

[THE_V01DL0RD]

What are the odds of having 3 Y-Perms in a row?

 

[Davian]

Or what is the odd of having a 3x3 step skip on 4x4?

 

[ProStar]

Or what is the odd of having a 3x3 step skip on 4x4?

Rounded to 20 decimal places, 0

 

[Jorian Meeuse]

Or what is the odd of having a 3x3 step skip on 4x4?

1 in 519024039293878272000

What are the odds of having 3 Y-Perms in a row?

1 in 5832

 

[Davian]

And how about a V skip in pyraminx

 

[Deleted member 74176]

Does anybody know the odds of a PBL skip on square-1?