Probability Thread

Discussion in 'Puzzle Theory' started by CubesOfTheWorld, Apr 8, 2010.

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  1. FastCubeMaster

    FastCubeMaster Member

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    O, definitely makes sense but seems a bit weird for me cos a few cases keep occuring, more than others. I guess it's just chance
     
  2. given a 2 random edges and 2 random corners being swapped on the cube

    what's the probability of a least one corner and edge being adjacent?

    examples of an adj case
    R' F R U2 r2 F r U' r U2 (UFR and UR are next to each other)

    non adj

    u R' U' R' F R2 U' R' U' R U R' F' R E
     
  3. Cale S

    Cale S Member

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    The corner pieces can be adjacent (3/7 chance), diagonal on a layer (3/7 chance), or on opposite corners of the cube (1/7 chance)

    Finding probability they aren't adjacent:

    If the corners are adjacent, first edge has a 7/12 chance, and second has 6/11 chance

    If the corners are diagonal, first edge has 6/12 = 1/2 chance, second has 5/11

    If the corners are completely opposite, the numbers are the same as diagonal

    1 - ((3/7)(7/12)(6/11) + (3/7)(1/2)(5/11) + (1/7)(1/2)(5/11))

    73.38% chance of having adjacent corner and edge
     
    Daniel Lin likes this.
  4. ok 1 more question

    how many total 2e2c cases are there if you count U/D, F/B, and L/R mirrors as the same?

    so that
    R U R' F' R U R' U' R' F R2 U' R' U'=
    R' D' R F R' D' R D R F' R2 D R D=
    L D L' F' L D L' D' L' F L2 D' L' D'

    can someone do the burnsides lemma thingy?
     
  5. xyzzy

    xyzzy Member

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    Dec 24, 2015
    Identity symmetry: $$(\binom82\cdot3)\cdot(\binom{12}2\cdot2)=11088$$
    Mirror on one axis: $$(4\cdot3)\cdot(12+8)=240$$ (12 for edges in the slice preserved under mirroring, 8 for edges outside of that slice)
    Mirror on two axes: $$(4)\cdot(12)=48$$
    Mirror on all axes: $$(4\cdot3)\cdot(12/2\cdot2)=144$$
    Burnside's lemma magic: $$\frac18(11088+3\cdot240+3\cdot48+144)=1512$$
     
    Daniel Lin likes this.
  6. Herbert Kociemba

    Herbert Kociemba Member

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    Is there a particular reason that e.g. F U F' L' F U F' U' F' L F2 U' F' U' should be counted as different? This belongs to a 90° rotation which cannot be generated by the three mirrors.
     
    Last edited: May 30, 2017
  7. Abram Lookadoo

    Abram Lookadoo Member

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    Jun 12, 2017
    can someone find the probability of...

    any 1x1x3 block formed after scramble
    being 1 move away from a 1x1x3 block after scramble (in axis turn metric)
    having at least one F2L pair anywhere on the cube after making a 2x2x3 block
    having both F2L pairs anywhere on the cube after making a 2x2x3 block
    2gr skip with phasing (and average movecount if able(htm))
    2gr skip with 2 corners oriented (and average movecount if able(htm))
     
    Last edited: Jun 25, 2017
  8. WACWCA

    WACWCA Member

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    Probability of solved 2x2 face, any kind
     
  9. Liquorice

    Liquorice Member

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    Feb 14, 2016
    What is the probability of parity in 3x3 blindfolded?

    What is the exact probability of getting an edges memo of length 11 (11 edges need to be solved, buffer does not count)? If buffer piece is solved, it adds 1 to the length. There may be a permuted flipped edge and a solved edge (11 +1 -1 = 11).
    Length 12 (one edge is flipped and permuted or two edges are flipped/permuted and one is solved etc.)? Length 1, 2, 3 ... 20, 21, 22?
    Corners memo length?
     
  10. Ronxu

    Ronxu Member

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    0.5
     
    Keroma12 likes this.
  11. Liquorice

    Liquorice Member

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    Feb 14, 2016
    Why? How do you know?
     
  12. Ronxu

    Ronxu Member

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    If the scramble has an odd number of quarter turns you have parity. This happens because a quarter turn does an odd number of both edge and corner 2-cycles. if you have an even number of quarter turns, then the total number of 2-cycles is even on both edges and corners and there's no parity. Both cases are equally likely.
     
  13. Probabilities of having a 3/4/5bld solve without parity or any cycle breaks.
     
  14. xyzzy

    xyzzy Member

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    For 3bld, it's around 3.97%:

    4bld is too hard to exactly calculate because it depends on how you orient the cube. Too lazy to calculate for 5bld now, but I think you can reuse the above calculation and replace 8 or 12 with 24 as appropriate for the wings, and for the centres you don't really need cycle breaks at all, I think.
     
  15. greentgoatgal

    greentgoatgal Member

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    Feb 6, 2017
    You are also more likely to get struck by lightning on your way to get the lottery ticket than you are to win the lottery.
     
  16. If you drove to the gas station each time to buy a lottery ticket for the expected number of tickets you would need to win the lottery, you would be killed in a car wreck 6 or 7 times before actually getting a winning ticket.
     
  17. What is the probability of an 4 move 2x2 solve?
    And what's the rarest and least rare pll?
     
  18. KAINOS

    KAINOS Member

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    Jun 27, 2016
    Somewhere in Earth
    For the first question check out this website: https://www.jaapsch.net/puzzles/cube2.htm
    The rarest PLLs are H and N perms, (probability=1/72) and other than E and Z (1/36) the rest of them have the same chance of happening(1/18). If you group variants as a single case the most common one would be G perm, though. (1/18*4=2/9)
     
  19. RubixKid

    RubixKid Member

    The chance of a last 4 centers skip on Skewb is 1/8

    I got 4 in a row once which is a 1/4096 probability!
     
  20. Cale S

    Cale S Member

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    it's 1/12
     

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