[Closed] Scrabble question, probability

less if you live in Wales

back pages of New Scientist do this sort of stuff even had a question about corrugations on gravel roads a few months ago...pretty dubious answers

It would depend on how many and which letters have been taken already.

From the point of view of statistics, AAEEIOU is just another word.

At the start of the game there are 100 tiles and 9 As, so the probability of getting an A is 9/100. Then the probability of getting the next A is 8/99. There are 12 Es left in the bag which now has 98 letters in it, so your probability is 12/97. The probabilties stack, so work out all those numbers and multiply them together and you get your probability of that pariticular word at the start of the game.

Some words are more probable than others because of the distribution of letters. The string of vowels is more likely than say QUETZAL (a type of bird) because obvs there are more vowels in the bag than Qs and Zs.

Where it gets beyond my A-Level maths is how to calculate the probability of it happening at any point during a game. It clearly depends what has happened in the game before, and how many words have been played. And this remarkable event is notable at any point in the game, so you need to stack all the probabilities which either diminish as vowel-heavy words get played, or increase as consonant-heavy words get played. There may be a way to create a model averaging out the course of a game and stacking those probabilities, but I have no idea off the top of my head and I CBA to consider it in depth 🙂

less if you live in Wales

Seven vowels in the Welsh alphabet. The fact you don't know what they are doesn't constitute anything weird about Welsh.

Han Solo wouldn't want to know.

Seven vowels in the Welsh alphabet

my mini brain that doesn't know what they are reckons that means that pulling all the vowels in order is less...maybe my pointing out that defining vowels depends on location was wasted

Seven vowels in the Welsh alphabet

L,W,Y,L,W,L and L?

Thanks molgrips, that's about as much of an exposition as I hoped for, or would be able to deal with.

I often wrestled with a similar question regarding the National Lottery (and had many arguments about it in the pub with mates) - is it equally likely for the numbers coming out of the machine to be (in this order): 1, 2, 3, 4, 5, 6 as it would be any other random numbers such as 25, 4, 49, 46, 12, 31?

is it equally likely for the numbers coming out of the machine to be (in this order): 1, 2, 3, 4, 5, 6 as it would be any other random numbers such as 25, 4, 49, 46, 12, 31?

Yes, it is. Any given combination is very unlikely, that's why 1, 2, 3, 4, 5, 6 would be really surprising. If you predicted beforehand that the draw would be 25, 4, 49, 46, 12, 31, it would be just as surprising if that was the draw as if it was 1, 2, 3, 4, 5, 6. Trick is, you have to predict the numbers before the draw, not afterwards.

As long as 25, 4, 49, 46, 12, 31 had to come out in that order, then yes it's the same probability as 1, 2, 3, 4, 5, 6 in order.

Without knowing how far through the game you were, ie. what other tiles had already been drawn it would be impossible to say what the odds of that combination in that order were.

Probabilities of any particular "hand" in Scrabble are fairly complex to calculate because their is an uneven distribution of tiles to start with, and then it changes every time someone plays. The fact you only picked 5 and ended up with 7 vowels further complicates the issue (you must already have had 2 vowels which must themselves have been "in order" you then presumably had one choice to place your pieces to the left or right of them.

The result you got was incredibly unlikely - as is every individual draw of tiles (at least in the early stage of the game). In fact because vowels (and A,E,I in particular) are more common than other tiles your outcome is less surprising than other combinations that would likely go unnoticed.

AAEEIOU is just another word

It's not a word though, is it? If DK's letters said a random 7 letter word that would be more impressive. It's not even a particularly interesting pattern (other non-word patterns like ABCDEFG would be more interesting, for example).

As per the National Lottery Example, the chance of your precise combination occurring is admittedly very small, but no different to any other precise combination of letters (I'm ignoring Scrabble letter frequency for ease of argument, though again that increases the odds of your particular combo vs anything with a lower frequency letter in it). Google says there are 32909 seven letter English words, plus there must be a handful of other interesting combos like ABCDEFG, TUVWXYZ or acronyms like WYSIWYG.

So, OP, when you multiply your probability by the number of interesting combos (E.g. 33000), then multiply that by how many turns you have in each game, and again by how many games of scrabble you've played in your life... You get a probability that is not so unlikely. If you're another degree of separation away (as anyone reading this is), and you multiply the probability (you just calculated) by the number of people who might tell you in passing about a special Scrabble combination (on the radio, TV, books, internet, friends etc etc), then, well, I'm sorry I'm not very impressed.

P.S. yes I am fun at parties. Technically.

is it equally likely for the numbers coming out of the machine to be (in this order): 1, 2, 3, 4, 5, 6 as it would be any other random numbers such as 25, 4, 49, 46, 12, 31?

Equally likely but a much reduced jackpot for 1, 2, 3, 4, 5, 6.

This is 96 sets of Scrabble letters randomized. As you would expect, lots of 3 letter words, surprisingly long strings of vowels. "BEARMATES" is kinda an 8 letter word. That's the longest one I could see from a quick skim.

TMZRJESRDEDZYIFUIINAEEGYGZPVTMKDEGSAIGSEETOTIIHFIRBSITAOEQKDTACCOEAWSEOACSIBRIKLIQCHOONAODUAUAGHBTRITADEGSRESYODLOIIMMEOIICINZUISEVHGMOAERKNTREJNIARHOSNNSNUOVRIEUTOSEEGYRTEYIAYOAMRHOVXIEWUTJOIPFEALWNHNIOEWERGRWGEOEIINLEOIHEZEKRTBBAEAAAODAISEEEIRAEETRYITDIRLDWAEFIELEADJQIOHMROSNTURZEITUGXNEIGCRNYSMOLJNNEUNGNASYENEDXIIEBUANACDOBWTWELGBYSNGLROTURBITHIAWOSSETBEUUENUEQYNTUVNIOSSRADFGOVOWYXCOYHEOKESAIIRDTOOJTAIAUETZIRRDAAQDBOOERUSTVUEYIROIAPNENNGIERAAYRALXIOENWPRGHEASEDZAAOOECSBAIZQIONTOYPATIITODOMITTLMNXPEOZDRSBBXFEABYAAIDWAMLAUSTVAFEEDATCQOOOTCEBYMKNIENNLTIOARAOILBTNCDAVUEICOOATRERWADDERUEDMUCPETRGDPJAWEELEGEERGYDTNIELPYODMEPENTASEFWVFGCOTRRIELECWSGOECZQIDKTTUVAUAHERTNAATRTQIHSSRRMNPOAQNNBIIUITLEILAAUTLARNOEMAOUBIEEBPOONIILICATNASSEIUOEIHQAAEEOALUEEFUTIUAKBLCAVIRUHLIBROEXVIQIVGYUJDFLEERNAESMTTEIJDIAEELPRGSSEMPIUWYVDPTZEEITMTIUIRUGSTIMMFNRIGIOEAIAAAITWIIRRENDTEEZLCAFIETTLOIWCERCILEILOFLVAIANMAEOAFPMORMLEUNEROTOHOTZHTTEGIAMMNOUVGTIYHZLQEIBLGDAINEPRTATQEIAOEFOXONSANATEDTEGKDILUXTDIDEUGEZHAUEELYAANCLLAFYAEIEIETAUNDOTXOIGCFVINUIIVSEUSWFEESOELKCEENBIDUTELUSAAXNYUAGGAUTDEOWEARURIRONINEEETEYYEEAEALIAIEATHDYLAPNINUSSRNIONIIWNDOWERMESORHDTNAEIIEIUHETRONNBOTRFNNOLNOASSCMEOIWHIIPRHDSRSNACABPUVETSEVGUZRRLNAERDGOLRZDJOHMDVJAREOXPOZENOSSHNARLLOPRTDREDRLHEOIOTOOEOCNQAEEILGAESRVGNIEYIBZACLNXBRECFINSAHEGUHEDCUEELNOGALOAOYIAERAOASIBIAYTUDOHEREECPTCDITOARITOOTGNHIEEAYTICNLEILUDANOREALERHOCOATMACAIAESQTLETOEITOLWASBHGAPSSVNNSQFIQITIRFIOAIMEREROINAERGFELLRBESNMNAZRETREPADTERPUSAOREOANTLJPIIRTEMIBITTRYEIFATWROEIREUNIMOEELODRPQZAKAIDGRWMXEOPNOUNIGWLCNGYBLYNREELUAVRLUURSNAREEIFILEIZGOLIFGEEBETYDDGEMVWONWOGIZGVBREAVINUNOUHRDOTNEEPDPGANJDRATRTVUAEAHOEEETULODTAIICSSGTRAAOIASLRNLPTNNOGILUQEEGBQYEOEUROEERIESSIAAXOSTLITAPUYOHIQTVKDOPERTPDZOTOOBRDLRLEOFARPTRAOHASNOTUAIENXEINTEEODFPUIWZGFETIRKIABURNOEYUSPHECEAAIDAOFGEVUIEAEMRCSOEONJYATAGRDRAXEBUTLIAOGAAEVRXOEVIDCKUTEEUIIBEALSLEAISLEONDUNKREEEBTTFELWONIIOIXRUXSTAOTMEAOCTGRMUSAEAEUURAERTCNVCBCSRNRNFGCBOYIGVOOUDJORPPRDOFDYENECLBMIVEEHAIDYNIDROEETYEIXSRENTOFFZUIIOPEUITNRELHEOAMOSJUFCOIIDHRGTGNGIICJBEYEOJEPATSREIIFSERMOLRCAITIIREUREWGOADHTGGSWPNRTIOLOSOYDROLGDADGGBOGNOETEATSYDRLOYVORPOANEHITFYOTEIRWWZRNOFOFEIWAOVLIARIROTEFILUMETERAOQNTRNIWTTNAGOXYHNUTNKUORROEASITRZSAILEGDDHEAILLNTCPTGURVLONELCOLNIEEAOOEFMVLASIANIAXAEYAAISIAZENIPBONODHEHIOIROGNBNTABEOPOAXFAUREWMUAKRARIAEAUTTBAGRWOOIILNJEDUEAWNYQSVOAAAIHLTGRIEMNNICUTDGSCREEMLELRDOEEOAIEEFIPOGDJIINRINRPEHVOEEAFNNLERFRRBUUGEGGPOBARIVCISIAMQTOETTPIAHRENGVWCFODNJMEXIUBAROISLIMFNIIILWEKTNETETASIUGKDYANSDINOTNIONIRGOTWAESTIASNPEELROEATEAUTVALEOJAEXNXEOENLTLPLDFEGEEENWTIRAMOECSSTINRWIAEAEEBNSSUGMTCNSAREOOOESCGTGQEEIJGHEEIERTINUULHOPAAATJXTITTDMULBOAENGDSCRLRMINPESEDRAGRAENEMROOTAPAVMARAUNNILSIEUJEAOMDESIWAGTWSSANZCOMNOZDLAQEDNCIUEEGRIMPLGAWANNSSDENMOSIRIUIYBTUVITOAGUAOLVOAUWNVFTWNRNLGITIAEXENTAEIISVSNENEHTESTEZREOROVIOISEHIAAANGEWYEFKTKASYOJEOFAIABCIIWHARDVEVISZSDETNLGTTOKCARTIDGPAIEIDIAFZVKEGLDLTLYASIEEEAENNAEKGMNTILRONOUCOUVTNIAIYEUIQZINRMBELRMOESEELOHICDAETIPASEUIUURTDIAJJORRISRTQIYELNLADYEBWSLAKNOGVRUAINATUINQAIRIRMRYQDHQTIIOIDSQUWIIGEARFUTHKBBTTIVEUSINUIDEEIDALIXIRRIWIMHUCOIRFWQNUAAAZATAGEKTNAYCINODAXIUFONOPGEOAORCDRROQLORDEIWNSJVITEAAUSOAJOUAUMBOLPRBAEOIFENBIHUELIPVCMPTOTDNOUHIITUDLAZOEIUDAPACBEIEKJONAIDAOSTEMIOGUPYOEGOURTHOIPTRUOUEYACEEPHEEDOIODVDNTOSMMOZBOTEEQDPMELDICICDOETOOIKLGTAOOITPEYIERYCLLWTLXFPBASEAGHJYWONNAEECADKIIRDOAIEISAREEHNNRLEAUOEGHHHTRGEOQNUAPLIEWPURPEZNRLATNPTPLTBCFDAYVRFIARYAXIARNNDRGEIYLMITTEGLDEUEIEEDOIEYIWGATIQYOEOOEMJZEAERCYGNILQTWNAQUAAEWUBAIFMASTMNOYPENEOIEOUIMSRXNBUIHEWRVDDYNMLPLVAZBQSIECLRCSEENEZNSMIIOGTINLMAACNAGISSYIADAGNSEIRBAEEANGTPEYSETRSGCOEYKSCWONFYFNYHRELTODNITIMASGVBOKMFTEAOPRSRNNMEHVOIIMOAARGSEIIARANITFLURAESONQICETKGBRFZTGVWEHTCAEOGANMARSIESEOHTTEEUROIILJIOEXTDIERNSFNEEBSHERJAUKETALRMJEEAEWAOXOABIOEKGQITNAIPUZNOATNAYIHAALWATAADMEGEKWIPORWIEAIEERDQAINSLXNLTNGAVIAXJBRGYRFNHAPAEANKCTFADXRPLBOEFRTIMPSUVJDOEIENWNHDIUWVUNTOAXCVMLINEVPSONNBESBLAMVGOEELXGSOANEMTEVBUVOFNNOEIBSEDDDXESOUARIEENIAFLAIREMHAIEAMTFGNDGURAOAHTEYSDERSQMIDRNODHEUECEDTPKBSFNFVBNEENAWAIINSSLARBTJASANWUSAERTBDTIDILSOTOSPZOMUTECTNILREAEEARAIPMMRQTSRRLKEIHONITVLADUTTAGPFNEETEEETYDILUZUKAIOJLNAUQUYETEEODGROEEUZIFUOIIPHBWINARQDTEECDOTLTNOCETIFEAOAZAISPGLEIURRSEIETOANSNDAUVABNOFUEEULLEJRSTNIUIIALIAEQIENCIGUIIOADRMIBGCHONIDCDOTKOEAUTEAMFTAOTOHAHDTREERIFOUEELTARANTTVBYULETMMOKWTSUAOIRQTAETDOTTHMNJIHBIPEVINIGVIIEAQANAIDRETOTIOEOXEKIXEQNEGUNIBORIAAADGIFIOEAOOUBEARMATESEISEFOIAFGACNNESIOSEPGADUERBTILEYZSJBTMAUFAOHOPSEREDEIDMSUGIZREOOOOYMEHROIZYAJTATREUATAERXHSNHEVETAEABRTLNGSOOTAEMZEORTWXUYFGQIPIYGNQZEMBLDOIERRUAOTEEOCXTOURFHMYBWDREOVIEGWDOAATTTNUTSLIILALOAARWNAOAAANAIINTDGDELDGNRBOEDEYQLNDVRIRDYEIZUDNTOTOYKTSLONDXAEBMRRJDADTLUANAINAIVBUDXULIEEDASOSIZKYCEIHEANTMEISYWTIGBFSELTFRKTOTNLIIGPEUATPGEQIESBBJLKEIILFTYNAUNVSRRVREKTEUOKIMNCMRRRHRRSRAIUZYIAMCREIEONOOREOURFCLEDOIWTEOHEBAIYWVSMLCIISTTNANTEBUOOACMERENHTJANDITMHCUGONAPGSSLOIRRVTDOAECHUAUEAWPTAIBLTBTGOSWBDLILETHIDAYDOVDRLAETDTKEOEEEKTUNOIEJESFEMJEEJTNEEAUTUKKWTTKOVARIEWDLNYAPUZOVBAXIXUINWSGJRUADIIITTQUWEVWPDEAA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I was hoping to see some amusing words or phrases in that list but I'm drawing a blank

I was hoping to see some amusing words or phrases in that list but I’m drawing a blank

Weirdly though 'didnhurt' appears 3 times.

And if you sort of squint and defocus a 3D image of pterodactyl appears

Some great posts here.

The result you got was incredibly unlikely – as is every individual draw of tiles (at least in the early stage of the game). In fact because vowels (and A,E,I in particular) are more common than other tiles your outcome is less surprising than other combinations that would likely go unnoticed.

This is a great observation. Drawing JXDFEIA is just as unlikely as AAEEIOU but you simply wouldn't care about the former.

So, OP, when you multiply your probability by the number of interesting combos (E.g. 33000), then multiply that by how many turns you have in each game, and again by how many games of scrabble you’ve played in your life… You get a probability that is not so unlikely.

Yes - you see millions of 'things' every day - car number plates, phone numbers, people's birthdays, names, etc etc etc, the chances of 'something' popping up that make you 'oh look at that' are quite good.

I like how Hols has snuck in 'KILLSQUIRRELS' a couple of times. Quality. 🙂

I'm surprised you weren't disqualified for vowel play.

the chances of ‘something’ popping up that make you ‘oh look at that’ are quite good.

0.63 in fact

This is 96 sets of Scrabble letters randomized.

I asked my computer to whiz through looking for words from its inbuilt dictionary. It found some proper nouns, which don't count for Scrabble playing of course, but still look impressive if you drew them out.

Anyway, here's the counts for words it found:

280 3 letter words
108 4 letter words
12 5 letter words
1 6 letter words

The six letter word is "realer" FWIW.

I'm not sure what this shows 🙂 other than it's fairly rare to draw meaningful combinations of letters in the right order once you start looking at longer words. OP's example is seven letters, and there's no seven-letter word in that particular 96-games-worth of letters.

0.63 in fact

Wow really? My mate was born in 1963! What a coincidence!

Yes, it is based on the premise that there are an infinite number of unnotable coincidences, which when expressed algebracally and maniuplated a bit ends up as a geomtric series which tends to 1/e. Therefore the probability of a notable one is 1 minus that, and 1- 1/e equals 0.63

And if you sort of squint and defocus a 3D image of pterodactyl appears

That, it turns out, is actually a possibility

I’m surprised you weren’t disqualified for vowel play.

🙂 Thanks - I shall use that joke when play is resumed.

Yes, it is based on the premise that there are an infinite number of unnotable coincidences, which when expressed algebracally and maniuplated a bit ends up as a geomtric series which tends to 1/e. Therefore the probability of a notable one is 1 minus that, and 1- 1/e equals 0.63

Interesting.