Can You KNOW that Your Lottery Ticket is a Loser?
Sure, and You Can Also Know that the Lottery is Unfair if Anyone Won
Lots of people think that you can’t know that a given lottery ticket is a losing ticket, before the winning number is revealed. If the ticket is in fact the winning ticket, then it’s of course pretty reasonable to hold that you can’t know it’s a loser, since it’s not a loser. The consensus opinion is different. It’s the claim that even if in fact your ticket is a loser, you can’t know that fact ahead of time, before the winning number has been publicized by lottery officials (even though the winning number has indeed already been determined).
Of course, you could know ahead of time that it’s a losing ticket if you had superhuman powers, such as reading the minds of the lottery officials, seeing into the future, or something comparable. Or, you could know it’s a losing ticket if a god told you so and you knew the god was omniscient and wasn’t deceiving you. It might seem that we can safely ignore silly possibilities like those. It turns out we can’t, for reasons I’ll get to below.
The consensus is that a normal person, regarding a normal lottery, under ordinary, realistic conditions, can’t know ahead of time that a given ticket is a loser.
I disagree, sort of. I have a complex thesis regarding lotteries, which might strike you as counterintuitive:
THESIS: If you know that a lottery is extreme, then before any announcement of the outcome of the lottery, you can know that if it’s announced that someone got the winning ticket, one of the following is true:
(i) The announcement that someone won is false.
(ii) Someone did win but something supernatural or superhuman was going on.
(iii) Someone did win but the lottery was fixed.
That is, you can know, before any announcement, that the outcome won’t be an ordinary case of someone winning a fair lottery, with no superhuman/supernatural stuff going on. Hence, you can deduce, for a given lottery ticket, that it is a loser provided there’s no supernatural/superhuman stuff going on and the lottery was fair.
What “extreme” amounts to:
Suppose a lottery is operated as follows. A random number generator used by the lottery commission chooses an ordered string of 10,000 digits from 0 to 9. That’s the one and only winning lottery ticket number. When you go to a store to buy yourself a lottery ticket, you have to specify the 10,000 digits for your ticket. You can either do it yourself or have the store’s random number generator choose the digits for you. Or, you can do a combination of the two, like when you choose your childhood phone number for the first few digits and let the store’s computer randomly generate the rest of the digits.
Hence, there are 1010,000 possible tickets total. That’s a ridiculous number of tickets. Physicists once estimated that there were 1080 atomic particles in the entire universe. Suppose they underestimated by a factor of a trillion x trillion x trillion (which is 1036). That’s a stupidly big mistake. If so, then there are 10116 particles. We’re still nowhere near 1010,000. 10116 is infinitesimal compared to 1010,000.
Suppose they underestimated the number of particles by a googol googol googol, where a googol is 10100. This would be the worst prediction of a finite number in history, no contest. Even so, we’re only up to 10380 or so, nowhere near 1010,000.
Someone might initially think that since 10,000 is roughly 3 times 380, 1010,000 must be roughly 3 times 10380. Nope. 1010,000 is 109620 times 10380.
Just to make things clearer, suppose the lottery tickets had just nine digits. Then the odds of a random one winning are a billion to one. If it had twelve digits, it’s a trillion to one. If it has 100 digits, it’s a googol to one. We’re nowhere near 10,000 digits.
What about the response that says “The 10,000-digit lottery is only about 1111 times as extreme as the 9-digit lottery, since 10,000 is about 1111 times 9”?
No. The 10,000-digit lottery is not 1111 times as extreme as the 9-digit lottery. It’s 109991 times as extreme, not 1111 times as extreme. That is, your odds in the 10,000-digit case are 109991 times worse than they are in the 9-digit case.
In order to get a vivid sense of the numbers involved, it helps to visually imagine what’s going on.
Imagine you’re watching a video recording of Fred choosing his lottery ticket’s numbers at a 7-11 store. Perhaps it’s footage from the store’s security camera. You watch him choose the first digit, then the second, then the third, then the fourth, and so on, up to, say, twenty digits. After those twenty digits, he has to do one of several things.
Have the store’s random number generator choose the remaining 9980 digits.
Have his own computer, that he accesses from the store (perhaps via his phone or small laptop he brought with him), choose the remaining 9980 digits.
Recite the remaining digits himself, or have a friend assist him. He could do this in a reasonable amount of time if he used a pattern of some sort, such as “Just do ‘142857’ over and over again, for the other 9980 digits”.
Here’s the winning number (and yes: it’s 10,000 digits):
97635399469181058913517232628329286179659748049083547521341928419062828624008558082014059594799572009892069809166402208543728018493241591821064258194960334619855901463351662561723321592726323858090878142953140411410618351308764045389757262340329862912105337708215961179960316716188430269010855161911835073710560210432701363776602665411089625384609463175378689125913248927504032362089842843819140250873740711177781526534433552075240584462212666371948017220404579192019546775916117904202554315223923577906321101279280575862622993001984211430093814156884655256044074844848489938301738238850226940110797673131738767662190343699110228965824670106412121125700276714563315015895539074607843633261912373787980687930654336312428843508221666676578209455033989505313098433115945269158566244212983727452139786532931542439873743306942321911005899822311368872946640452783616112819569795453990398218958008084853741003243406357199907413641579637239425234519413684667533673851741659072456724608084844977072756238047911396031522347563186293105449290250277855638640957329761340812573127093978786655426066145767445441645147220432190774533608157234156322988408590850223248009272229425882928856065165694546646773997023144297369906070391311101090130283402383532320151287329024730186506737422018250283172978982400415935810750365083359092844112074994248838596996086174719050902585742776505306418001732764461192868105783109889422050478592126845723928683317075356337247526317387014499199084886784369867180131435028112689477659445998922299128250513411708730030125348389408390987269057817169886103302539517508239520080417604476631482470479567071188327161346895128851615231930763471261374768337705991199669953444383765975027481508992923578021173740673555568979087899841971992080709316198741470726160845793538325496645900126652888881049770554926729473295046949376878769132479871316475150007829379087302867497275106269595630763525675589736385516113731652947387170841288594794857050779169708467440971242889319112258759340346432257547205648953872865218137430657850712931505180319783196269784769318597041996108001486327883672084181536591282343012128442409358110116078377601725493204614321234832493948567145013169917168997063255170076758425172245443101677367180397836342793153631740689288799826643299629819103352909975914606905444798988795095019357982962413192520214664202001621720066614928217445275477868692191749928178814159712607165806462708141792988824698692143911126204254095977105785775839883135313731044936930773038770612441628047138201591969865161310376603149212026292246956641418446088934834873898012733058745140730486819377889917158751540523188084639271420218020673751129175931886283734173990155735036828978081935607111250605826864611433508352755236045030822167003902055235747116177197637102263092621316201455764369400357763652635979872140682002969663611539702969417496492673763771337037029100125757719508830023073296052268274472913109581865375658710325846517107498485228497403511202779979891388589552949579371552419324686352854907034081891083036851756582114972187695686328012183445958433417518420958407125182496200972812469787274564681579665795449628536622817385578095500151982745497805819720508417752532491634336163765196033409163835350927763549950158509247287997179963822909159281809346051920421429366189186559919274214456137849158956539226546934996809533410017687306030755142653458182109227723973314200312866401322149355070355550270460177804734288356941267185607901305310100860039276776342438557065498425346614915526398236485439680694454249441207924133525537423671575626393646691031322008151505280819642015301249905111487936911381791030243590310831494242647009955730705027740143546736081589061347096095972345786232144647045020951058025840618111730584546126710282930272923962676413854723619454181640147751605006861176589484320119320755169355888003662981903478524601078370532260783468292330754208492030050878885689502481052819192359534539740405148836226636704152076309217373632149814620311217255815000870995956517882099965118210863284641097666372510167463482591251621453016754994161311004557785038033399264650227051333220817472660872824927041818866681091149501434456033492247031273374719898536200070189586146188897741660620784788843712206081650722570316165177201877782495686025138784127792650976199809911231173905761581491090090321650013313010753477575173552713067419958468179954662592488036214963337158593381448607451193136674230605750970105457006366333694281940395341184912893872548760834663407076962055425972596013676311514462051299815873514475079695849414486070145861071943405057569704065528242924565303007516902946048872192770302310316584851268813563854679735509947376724939769062375001640828360678134996319744872168728433732664794139884369028893743706702213508135740394470343497231060058623819143550868508891072827175758935433562133708285893445141644458569341062905003499719956699063537428207094214761099406521723042269530644223610061461252296780296260047494581512314975711320089294096544538532829630717123899416199539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
Go back to the store’s video recording of Fred. You’re watching the video recording after the winning number has been announced, so you know the winning number. You watch Fred choose the first digit correctly, then the second, then the third, then the fourth, and so on, up to, say, twenty digits. It seems insane that he’s getting each digit exactly right. After the first twenty, he does what almost everyone does: he has the store computer randomly choose the remaining 9,980 digits. And you see it choose the exact right digits in the exact right order.
You would know—right???—that it can’t be a coincidence that Fred and the store’s computer got all 10,000 digits right. Sure, he could, just by luck, get the first 5 or 7 or even 9 right. Not impossible. But seeing him or the store’s computer get all 10,000 in a row exactly right means that something’s amiss:
He somehow knew ahead of time the right 10,000 digits. He’s a god, with supernatural abilities. Or he’s a human with the superhuman ability to tell the future. Or he knew someone with those abilities who told him the right digits.
Or he didn’t know ahead of time but someone god-like was behind the scenes, subtly manipulating both Fred and the store’s computer without anyone knowing about it.
The store computer is somehow appropriately connected to the so-called random number generator that the lottery commission uses.
Or something else equally sketchy is going on.
There’s just no way that he and the store computer got the right 10,000 digits without something weird going on. That’s what you can know.
*****
Clearly, if the lottery uses just three digits, so there are only 1000 total ticket numbers, then you can’t know ahead of time that a given ticket is a loser. And when you hear that someone won, you can’t know that something fishy was going on.
So, you might ask, when does a lottery get big enough to make the thesis true?
That’s the much harder question, in my opinion!
Very interesting! I suppose one thing that makes it slightly counterintuitive is that you never stipulate that the ticket is in fact not winning. It should obviously barely make a difference given the numbers involved but I still think it helps make the point more intuitive.