Probability Thread

I was solving with Roux and after my first two blocks all I needed to do was M2 U2 M2 U2 and the cube was solved. It was a hand scramble and I wasn't timing but it definitely would have been a PB if I was. What is the probability of this case happening? Is is more or less likely that a full last layer skip in CFOP?
 
GiraffeCubing said:
I was solving with Roux and after my first two blocks all I needed to do was M2 U2 M2 U2 and the cube was solved. It was a hand scramble and I wasn't timing but it definitely would have been a PB if I was. What is the probability of this case happening? Is is more or less likely that a full last layer skip in CFOP?
Click to expand...

Well, basically you skipped CMLL and the first two L6E parts:

4! (corner permutation)
/4 (AUF)
*3^3 (corner orientation)
*2^5 (edge orientation)
*6*5 (placing UL and UR edge)
= 1 in every 155520 solves, which is exactly 10 times more unlikely than a LL-skip
 
What is the probability of a solved cross on a 3x3 scramble? (For Stefan's sake, what is the probability of all edges of at least one layer being correctly oriented and permuted). Also, what is the probability of a cross's pieces being correct relative to each other but a move is needed to correct the centers? I got two solved crosses on scrambles in the past two days, one was completely solved and the other was off by a D move (cross on D).
 
rybaby said:
What is the probability of a solved cross on a 3x3 scramble? (For Stefan's sake, what is the probability of all edges of at least one layer being correctly oriented and permuted). Also, what is the probability of a cross's pieces being correct relative to each other but a move is needed to correct the centers? I got two solved crosses on scrambles in the past two days, one was completely solved and the other was off by a D move (cross on D).
Click to expand...

i think its 7920 and 31680, but i'm not too good with probability.
 
What's the probability that a LL case is able to be AUFed into a 3 corner, 3 edge cycle? Of course G perms are such an example. Another is M U R U R' U' R' F R F' M' U'
 
guysensei1 said:
What's the probability that a LL case is able to be AUFed into a 3 corner, 3 edge cycle? Of course G perms are such an example. Another is M U R U R' U' R' F R F' M' U'
Click to expand...

I'm really bad at this stuff, so don't take my word for it

One edge and one corner solved

the three remaining corners can be in 2 different permutations for it to be a 3cycle (the third way they would be permuted correctly)
the three remaining edges can be in 2 different permutations for it to be a 3cycle (the third way they would be permuted correctly)
corners can be in 3^3 orientations, of which only a third is reachable with regular turns (so 3^2=9)
edges can be in 2^3 possible orientations, of which only half are reachable with regular turns (so 2^2=4)

2*2*9*4=144

the probability is 144/15552?


At this point of writing this I realize that I didn't calculate for some cases where you might be able to AUF it in two different ways so I don't know (I don't even know if that matters or not)

E: Should I multiply the 144 by 16? It wouldn't be the same corner and edge every time

2304/15552, or 4/27?
 
Last edited:
Username said:
I'm really bad at this stuff, so don't take my word for it

One edge and one corner solved

the three remaining corners can be in 2 different permutations for it to be a 3cycle (the third way they would be permuted correctly)
the three remaining edges can be in 2 different permutations for it to be a 3cycle (the third way they would be permuted correctly)
corners can be in 3^3 orientations, of which only a third is reachable with regular turns (so 3^2=9)
edges can be in 2^3 possible orientations, of which only half are reachable with regular turns (so 2^2=4)

2*2*9*4=144

the probability is 144/15552?
Click to expand...
No. 15552 is the total cases mod AUF. But you need to consider all 4 possible fixed corners, and all 4 possible fixed edges.




Username said:
Should I multiply the 144 by 16? It wouldn't be the same corner and edge every time


2304/15552, or 4/27?
Click to expand...
Yes, but but this is still not the final answer.

Username said:
At this point of writing this I realize that I didn't calculate for some cases where you might be able to AUF it in two different ways so I don't know (I don't even know if that matters or not)
Click to expand...
Yes, it matters. You've overcounted the number of cases because of this.

I've done a brute force enumeration using GAP. The result is that the actual probability is 11/81. There are 1920 cases out of 15552 where there is a single AUF case that satisfies the problem, and 192 cases where there are 2 AUF cases satisfying the problem. (1920 + 192)/15552 = 11/81.

GAP code:

Code: 
U := (1,3,9,7)(2,6,8,4)(10,37,28,19)(11,38,29,20)(12,39,30,21);
D := (16,25,34,43)(17,26,35,44)(18,27,36,45)(46,48,54,52)(47,51,53,49);
L := (1,19,46,45)(4,22,49,42)(7,25,52,39)(10,12,18,16)(11,15,17,13);
R := (3,43,48,21)(6,40,51,24)(9,37,54,27)(28,30,36,34)(29,33,35,31);
F := (7,28,48,18)(8,31,47,15)(9,34,46,12)(19,21,27,25)(20,24,26,22);
B := (1,16,54,30)(2,13,53,33)(3,10,52,36)(37,39,45,43)(38,42,44,40);
 
U2 := U*U;
U3 := U2*U;
D2 := D*D;
D3 := D2*D;
L2 := L*L;
L3 := L2*L;
R2 := R*R;
R3 := R2*R;
F2 := F*F;
F3 := F2*F;
B2 := B*B;
B3 := B2*B;

Aperm := R*B3*R*F2*R3*B*R*F2*R2;
Jperm := B2*L*U*L3*B2*R*D3*R*D*R2;
fururf := F*U*R*U3*R3*F3;
GPLL := Group (U, Aperm, Jperm);
GLL := Group (U, Jperm, fururf);

FaceletCycleLen := function (g, n)
  local i, j;
  i := n;
  j := 1;
  while i^g <> n do
    i := i^g;
    j := j + 1;
  od;
  return j;
end;

CubieCycleLen := function (g, n)
  local i, j, cmap;
  cmap := [
   1, 2, 3, 4, 5, 6, 7, 8, 9,
   1, 4, 7, 13, 14, 15, 52, 49, 46,
   7, 8, 9, 15, 23, 24, 46, 47, 48,
   9, 6, 3, 24, 32, 33, 48, 51, 54,
   3, 2, 1, 33, 41, 13, 54, 53, 52,
   46, 47, 48, 49, 50, 51, 52, 53, 54
  ];
  i := cmap[n];
  j := 1;
  while cmap[i^g] <> cmap[n] do
    i := cmap[i^g];
    j := j + 1;
  od;
  return j;
end;

Check33 := function (g)
  local ok, c1, c3, e2, e4;
  ok := false;
  c1 := CubieCycleLen (g, 1);
  if (c1 = 3) and (FaceletCycleLen (g, 1) = 3) then
    ok := true;
  else
    c3 := CubieCycleLen (g, 3);
    if (c3 = 3) and (FaceletCycleLen (g, 3) = 3) then
      ok := true;
    fi;
  fi;
  if ok then
    ok := false;
    e2 := CubieCycleLen (g, 2);
    if (e2 = 3) and (FaceletCycleLen (g, 2) = 3) then
      ok := true;
    else
      e4 := CubieCycleLen (g, 4);
      if (e4 = 3) and (FaceletCycleLen (g, 4) = 3) then
        ok := true;
      fi;
    fi;
  fi;
  return ok;
end;

Check33All := function (G)
  local g, g2, g3, g4, h, j, c, Lused, Lout, Lc, Le, Lx;
  Lused := [];
  Lout := [];
  for g in G do
    if not (g in Lused) then
      h := g;
      j := 1;
      Lx := [];
      c := 0;
      while j <= 4 do
        if Check33 (h) then
          c := c + 1;
        fi;
        Add (Lused, h);
        h := h*U;
        j := j + 1;
      od;
      Add (Lout, c);
    fi;
  od;
  return Lout;
end;

result := Check33All (GLL);;
nx := Size (result);
n1 := Size (Positions (result, 1));
n2 := Size (Positions (result, 2));
(n1+n2)/nx;
 
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