Probability Thread

cubernya · May 3, 2012

kinch2002 said:
Here are the solutions. It's a little off-topic to be talking about whether this is Fridrich or not, so whatever. I'd say everything is standard Fridrich apart from from Jimmy starting with 2x2 block and that István kind of switched from green to yellow F2L, while cancelling a pair into the OLL. Reconstructions are posted by others from other threads

Jimmy:
Scramble : U L2 U' R2 B2 U L2 D2 R2 F2 R2 F L F' D2 L' U B' D R2 F'
Solution : U L U R2 U2 (bloc 222)
B D' R2 D' B R' (cross + second F2L)
B2 U' B U (3rd F2L)
D B2 D2 B' D2 B' D' (4th F2L + LL skip)

István:
scramble: D'R2DL2F2DLR2B'D2U2L2DF'L2UL2FU
Xcross on green: FD'FL'BR2
2nd f2l: RBR'B2DBD' (12 move with R2 R cancel)
and from here he should've done: U'BU'LU2L' (18 moves total, beaten the scramble by 1 move!)
but he did something like: U2BLU'LF'L'FU'L' (22 moves total)

I do remember seeing that alternate ending on another thread. IIRC the 18 turns is optimal. Of course, the threads are somewhere on the forums, but I don't have them bookmarked and have no reason to search for them ATM

kinch2002 · May 3, 2012

kinch2002 said:
Reconstructions are posted by others from other thread

theZcuber said:
I do remember seeing that alternate ending on another thread. IIRC the 18 turns is optimal. Of course, the threads are somewhere on the forums, but I don't have them bookmarked and have no reason to search for them ATM

As you can see, I said that they were not my own work. I found the original posts and copied the text into my post.

cubernya · May 3, 2012

kinch2002 said:
As you can see, I said that they were not my own work. I found the original posts and copied the text into my post.

:fp on my part

Eazoon · May 3, 2012

Rpotts said:
The fewest moves record was not achieved with Fridrich. Faz's fastest reconstructed solves have 9+ TPS, but he can turn substantially faster in bursts.

A solve that directly follows the Fridrich method without skipping any steps can't really be less than 28 moves HTM.

1 moves cross
4 3 move pairs
F RUR'U' F'
A/U perm.

1 + 12 + 6 + 9 = 28

If he gets really lucky, he could get a LL skip and/or f2l pair skip.

Julian · May 3, 2012

Eazoon said:
If he gets really lucky, he could get a LL skip and/or f2l pair skip.

Rpotts said:
A solve that directly follows the Fridrich method without skipping any steps...

yeah

Eazoon · May 3, 2012

Julian said:
**

oops.....

cuBerBruce · May 3, 2012

mande said:
Probability of at least 1 CE pair formed in a scramble?

I think mande has asked essentially the same thing once before.

It seems to me that the (approximate) answer to this must have been discussed previously, but so far I haven't found it. The exact answer appears to me to be rather complicated to calculate.

Using GAP, I computed the distribution of the number of CE pairs for ten million random positions. I also did this a 2nd time. Here are the results.

Code:

pairs  1st distr    2nd distr

  0      3684459      3686510
  1      3671274      3671462
  2      1837467      1835746
  3       613647       613283
  4       155370       155277
  5        31599        31464
  6         5318         5398
  7          745          752
  8          109           99
  9           11            9
 10            1            0
>=11           0            0

This suggests a probability of at least 1 CE pair of somewhere around 0.631.

Note that the probability of no pairs is close to 1/e. Why?

ThomasJE · May 4, 2012

aronpm said:
2.777... = 278% chance

On average, if there are 100 US Nationals, 278 of the US Nationals will have someone get a LL skip

Exactly. So it's virtually guaranteed. Even though I forgot about later rounds and 5 solves per average :fp. If I had remembered that, it would have been around 2000%

aronpm · May 4, 2012

ThomasJE said:
Exactly. So it's virtually guaranteed. Even though I forgot about later rounds and 5 solves per average :fp. If I had remembered that, it would have been around 2000%

NO
ARGHSDIBGERWNNASCNWG

PROBABILITY DOESNT WORK HOW YOU THINK IT WORKS

mr. giggums · May 4, 2012

ThomasJE said:
Exactly. So it's virtually guaranteed. Even though I forgot about later rounds and 5 solves per average :fp. If I had remembered that, it would have been around 2000%

Flipping a coin has a 50% chance. According to your logic if I flip a coin twice (.5 * 2 = 1 or 100%) I should always get a least one head. Try it this doesn't happen.

Noahaha · May 4, 2012

ThomasJE said:
Exactly. So it's virtually guaranteed. Even though I forgot about later rounds and 5 solves per average :fp. If I had remembered that, it would have been around 2000%

The problem is that probabilities are multiplied, not added. You need to say that the odds someone does NOT get a PLL skip are 71/72, and then the odds that in 200 solves everyone simultaneously does NOT get a PLL skip are (71/72)^200, and then the odds that there is a PLL skip will be:
1-(71/72)^200

Of course, 200 should be replaced with the actual number of solves. No probability can be greater than 1 or less than 0. Hope this helps.

cubernya · May 28, 2012

Odds of a solved layer on a scrambled 2x2?

ben1996123 · May 28, 2012

theZcuber said:
Odds of a solved layer on a scrambled 2x2?

It's been asked and answered in this thread before.

Nico1 · May 30, 2012

The probability of this is 44,497,945,755,648,000. You are more likely to win the lottery every single time it plays for the rest of your life than have a 3x3 skip!

ben1996123 · May 30, 2012

Nico1 said:
The probability of this is 44,497,945,755,648,000. You are more likely to win the lottery every single time it plays for the rest of your life than have a 3x3 skip!

No.

Lets assume I live for another 65 years. Lottery is played every week, so its played about 52*65 times.

Probability of winning lottery once is \( \frac{1}{13983816} \)

Probability of winning 52*65 = 3380 times in a row is \( \frac{1}{13983816^{3380}} \) =

1/164009452406338159331714771213929428624253674324941659470802651776681554840187006796572628401775075307189357509625995819415207408822885984939736188109682919867194426997559643943062618854258586740030738248874750751953504145003904690471593586795283496308016067623372140905975350264224275587856583684491219825425878174138734180417584191837833740722285690442068561619312026179657277736921512910086146244646720762445271315245749056347841980771609497037074447619269028512649949481675547810940639682080095186223158864302097383795918791198053686984028865297661101244336397607050684983862565814174994529307299733592553360963078272140830026551530801463630704984040731800391376872066139800152998030170347868128249596234722618448569337777675204993323091994793337555152638680456629927862885367578597602049385634016634215830661107078462148174649990718231587018649267462896403284324120745453457114446581757433956274825480462527813177102071459058476229016447977539052041827014032218019805778025489170977912921936115792781831769843209266723743156642646546838047356397407692975402108877797937607444029194249761599654192012919659206068842674293170815244263107055238203319037172027974986364897937089617484234005771281671228864920651244774273840358306621663102521800678882999790040623878721721997029305368239924331652353364590612101776499057981391535063229543400561741138399937820213543524750384657888142235519516623321266136678239360415123589111225538715037344412725603081106174763280000548518148363483266890307488994513831444868870897772106307709171806254613617425380494400738852291591537104570833573120255617217157496213736793019058375661648101277926991318474920463847125863730375351676842828373947029305961326589798194455295317058286951439789469713471522770645611530479360089112092499959009341477548210855813004897265364738786944306801239554146587919355973223338600763172030582294944372786896480941614623179727487522896340937427507870550687599204425151079914265014825523577327104339922347877887496322779216391122711853374302516418661763269009348446926825270365901515232186531706731575089843790972441617495784313414277650510202097553525325605294798183261639522565770933039813046882556062764180982191585907160307280199311660977746578118995450323174565589875007245131919159460053500536488234899189040289165682838928900082592287882873841018993166360042130976849251885745742753052808576807323214795275850512869554808063869863666085310154400349213182696208136928482524288016147544184530868983051345869752906564179626677760063802394667621206760547847680442983389283936454927540707550398784013057459861330218334268616510951648482475284023919266785751792442457531454143691633142428604217412352814522943970467897702722659408225757743653552767106972520409275384812573014023122612029931444715309834996467794700702112916714121068870934358723797037506447148412001363694417877282683146559352041870521083871508365804986535304588554166310169738365410317333099060189583757117400052813506400091464580309513488046434607822853291729887016233149540578850696711530693000914091947219475679391826127590855753100725265688083722285475682426976381641590521974101059648478549785061890935302100281113960717401326017953862437106328716410296295747225307759692707760658799677060828877421436482481593397503640199999835292271210228495429528256386440058093856066819152509406024468307197741943986865524104909275365212353321675520930248629053945942441808922008887610495318730944778859453688858775377632529478150466470081143672769588628536744473140718726995761603692127879396015670593115093666875426875483438986928046046784614880602005249564045401744449859801192522195905152052402508341013136192208182922479229909663857254586020790150232990787727808703686036434761562680355657543420582093048600456823966055605664387672766234170006651085840349130910525625262727188142252524987093277999467167679937716011910779015341385115563343248992569740636467349235428908095938992159070050355152789507659504920281605343133626667464226469188378940356726005730302957124485094889740322221815200331711834416292133065306674837823414852872857869333744661646223168028979386965453788393823460862865849458789402104424010492676606703052125877743644424486101835725271526457673898638815465860051058670960736589870341889110553904299144534452840233604774964301404689303737266250262794241241056992939345010540120119609935942017203840752111814904350378325633271437499599763842290039931298647895763369134883544250837992723070139196427057738558487154226741115257562033900793803735683080667462692161071424274879867169638616326005365818707712450046367996161722345305098402012575269475684905091992180047801880475853388104732495637913633090169477705770496657053373037053180386429988416475505498919617338941800216770372386824428274485562666952673207783342508616010658094012112820940160188502811220691733681045888911910568124939494900935530235163154518723168823821334234794445165925518264162426765609690420148364496260064056955229410400248756529161882481197232411002715891068565303112569627631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\( \approx 0 \)

CubeRoots · May 31, 2012

ben1996123 said:
No.

Lets assume I live for another 65 years. Lottery is played every week, so its played about 52*65 times.

Probability of winning lottery once is \( \frac{1}{13983816} \)

Probability of winning 52*65 = 3380 times in a row is \( \frac{1}{13983816^{3380}} \) =

1/164009452406338159331714771213929428624253674324941659470802651776681554840187006796572628401775075307189357509625995819415207408822885984939736188109682919867194426997559643943062618854258586740030738248874750751953504145003904690471593586795283496308016067623372140905975350264224275587856583684491219825425878174138734180417584191837833740722285690442068561619312026179657277736921512910086146244646720762445271315245749056347841980771609497037074447619269028512649949481675547810940639682080095186223158864302097383795918791198053686984028865297661101244336397607050684983862565814174994529307299733592553360963078272140830026551530801463630704984040731800391376872066139800152998030170347868128249596234722618448569337777675204993323091994793337555152638680456629927862885367578597602049385634016634215830661107078462148174649990718231587018649267462896403284324120745453457114446581757433956274825480462527813177102071459058476229016447977539052041827014032218019805778025489170977912921936115792781831769843209266723743156642646546838047356397407692975402108877797937607444029194249761599654192012919659206068842674293170815244263107055238203319037172027974986364897937089617484234005771281671228864920651244774273840358306621663102521800678882999790040623878721721997029305368239924331652353364590612101776499057981391535063229543400561741138399937820213543524750384657888142235519516623321266136678239360415123589111225538715037344412725603081106174763280000548518148363483266890307488994513831444868870897772106307709171806254613617425380494400738852291591537104570833573120255617217157496213736793019058375661648101277926991318474920463847125863730375351676842828373947029305961326589798194455295317058286951439789469713471522770645611530479360089112092499959009341477548210855813004897265364738786944306801239554146587919355973223338600763172030582294944372786896480941614623179727487522896340937427507870550687599204425151079914265014825523577327104339922347877887496322779216391122711853374302516418661763269009348446926825270365901515232186531706731575089843790972441617495784313414277650510202097553525325605294798183261639522565770933039813046882556062764180982191585907160307280199311660977746578118995450323174565589875007245131919159460053500536488234899189040289165682838928900082592287882873841018993166360042130976849251885745742753052808576807323214795275850512869554808063869863666085310154400349213182696208136928482524288016147544184530868983051345869752906564179626677760063802394667621206760547847680442983389283936454927540707550398784013057459861330218334268616510951648482475284023919266785751792442457531454143691633142428604217412352814522943970467897702722659408225757743653552767106972520409275384812573014023122612029931444715309834996467794700702112916714121068870934358723797037506447148412001363694417877282683146559352041870521083871508365804986535304588554166310169738365410317333099060189583757117400052813506400091464580309513488046434607822853291729887016233149540578850696711530693000914091947219475679391826127590855753100725265688083722285475682426976381641590521974101059648478549785061890935302100281113960717401326017953862437106328716410296295747225307759692707760658799677060828877421436482481593397503640199999835292271210228495429528256386440058093856066819152509406024468307197741943986865524104909275365212353321675520930248629053945942441808922008887610495318730944778859453688858775377632529478150466470081143672769588628536744473140718726995761603692127879396015670593115093666875426875483438986928046046784614880602005249564045401744449859801192522195905152052402508341013136192208182922479229909663857254586020790150232990787727808703686036434761562680355657543420582093048600456823966055605664387672766234170006651085840349130910525625262727188142252524987093277999467167679937716011910779015341385115563343248992569740636467349235428908095938992159070050355152789507659504920281605343133626667464226469188378940356726005730302957124485094889740322221815200331711834416292133065306674837823414852872857869333744661646223168028979386965453788393823460862865849458789402104424010492676606703052125877743644424486101835725271526457673898638815465860051058670960736589870341889110553904299144534452840233604774964301404689303737266250262794241241056992939345010540120119609935942017203840752111814904350378325633271437499599763842290039931298647895763369134883544250837992723070139196427057738558487154226741115257562033900793803735683080667462692161071424274879867169638616326005365818707712450046367996161722345305098402012575269475684905091992180047801880475853388104732495637913633090169477705770496657053373037053180386429988416475505498919617338941800216770372386824428274485562666952673207783342508616010658094012112820940160188502811220691733681045888911910568124939494900935530235163154518723168823821334234794445165925518264162426765609690420148364496260064056955229410400248756529161882481197232411002715891068565303112569627631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\( \approx 0 \)

some would say that 1/44497945755648000 \( \approx 0 \), too

ben1996123 · May 31, 2012

CubeRoots said:
some would say that 1/44497945755648000 \( \approx 0 \), too

\( \frac{1}{44497945755648000} \) is about \( 2.713*10^{24135} \) times larger than \( \frac{1}{13983816^{3380}} \) and \( 44497945755648000<43252003274489856000<13983816^{3380} \)

CubeRoots · May 31, 2012

ben1996123 said:
\( \frac{1}{44497945755648000} \) is about \( 2.713*10^{24135} \) times larger than \( \frac{1}{13983816^{3380}} \) and \( 44497945755648000<43252003274489856000<13983816^{3380} \)

yes but the difference between the two probabilities is 2.32...E(-20).

and the fact that it is 2.713...E24135 times larger is irrelevant when the probability of the event is so close to zero as you pointed out

ben1996123 · May 31, 2012

CubeRoots said:
yes but the difference between the two probabilities is 2.32...E(-20).

and the fact that it is 2.713...E24135 times larger is irrelevant when the probability of the event is so close to zero as you pointed out

whatever. it's not as though it's going to happen to anyone ever. probably.

CubeRoots · May 31, 2012

ben1996123 said:
whatever. it's not as though it's going to happen to anyone ever. probably.

lol. too true. nice probability pun there

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