[HN Gopher] Magician-turned-mathematician uncovers bias in coin ...
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Magician-turned-mathematician uncovers bias in coin flipping (2004)
Author : keithwhor
Score : 187 points
Date : 2020-12-24 13:56 UTC (9 hours ago)
(HTM) web link (news.stanford.edu)
(TXT) w3m dump (news.stanford.edu)
| jpcooper wrote:
| For those interested in games of chance, I've been playing around
| with Betfair's Exchange Hi Lo card game [1]. I came up with an
| algorithm to, given a game state, quickly compute the odds of all
| subsequent outcomes in polynomial time.
|
| Based on this algorithm, I've been building a bot [2] to
| integrate with Betfair's games API to play automatically for me.
|
| After finishing the first iteration, running it for a bit and
| losing a few Euros, I quickly realised that some of my
| assumptions as laid out in the README were wrong (I will be
| updating this soon). Namely, I wasn't accounting for Betfair's
| commission on winnings (silly me). I currently have an EU-based
| account, which has a 6.5 per cent commission on winnings. UK-
| based accounts have the minimum at 5 per cent. The commission has
| an effect on the cheapest profitable odds you can provide, which
| has an effect on whether people want to match bets at those odds.
|
| With this in mind, it quickly becomes clear that any unmatched
| bets you see in the market are far from the maximum profitable
| odds. This makes sense, and competing based on odds becomes
| impossible at this point. I am led to believe that the only way
| to make money is by having a UK-based account plus getting the
| bets to the market first. I have a few ideas on how to do this.
|
| If I can get the bets to match at odds that guarantee profit in
| the long-run assuming UK-based commission of 5 per cent, then I'm
| going to call it a day.
|
| Regardless, I'm not expecting to make too much money from this. I
| have a lot of free time right now (looking for a job), so if
| anyone wants to get into anything else available on Betfair
| together, then please get in touch at the email on my Github
| through [2]. My discrete probability and programming are okay,
| but it would be great to have someone who could share their
| statistics knowledge.
|
| [1] https://games.betfair.com/exchange-hi-lo/standard
|
| [2] https://github.com/jpcooper/betfair-exchange-hi-lo-odds
| teej wrote:
| ... you're aware of the adage "the house always wins" right?
| klinskyc wrote:
| Betfair is peer to peer, so the house just takes a rake
| (similar to poker). This means it is hypothetically possible
| for both you and the house to win.
| jpcooper wrote:
| On Betfair the house is you. You determine the odds. Let's
| assume that cards are dealt randomly. If I know the exact
| probabilities of every outcome I bet on, and I always bet
| with odds which have a margin over those probabilities, then
| I win in the long run.
| biztos wrote:
| Forgive my ignorance, but: why would anyone bet against
| you?
| coldtea wrote:
| The don't. They place bets. If they place bets on the
| opposite odds that you do on the same bet, then they play
| against you (and you're competing for that).
| asdf3331 wrote:
| for fun
| jpcooper wrote:
| Consider the case of betting with a traditional
| bookmaker. The bookmaker lays (bets against) an outcome.
| You back (bet for) the outcome. Each party is betting
| against the other.
| albedoa wrote:
| I'm a long-time poker player and recent political bettor.
| Gamblers routinely overestimate their edge _or_ don 't
| care whether they win. Often it's a mix of both.
|
| There are still many many many people who are willing to
| bet on the results of the US presidential election:
| https://www.thedailybeast.com/gamblers-bet-big-on-trump-
| and-...
| jpcooper wrote:
| What do you base your political bets on?
| albedoa wrote:
| For one-offs, picking the favorite is good enough for
| most players. Thy are just seeking entertainment and
| maybe to win a little money. For instance, in a race
| between two candidates priced at $0.70 and $0.30 (ranging
| from $0.01 to $0.99 before resolution), many players are
| happy to pick the favorite indicated by the market.
|
| Of course, if those prices are accurate, then they will
| lose money in the long-run after fees. For players who
| are betting more frequently and over a longer term, it's
| like any other speculation: You might assess the $0.30
| underdog to have a 40% chance of winning. She is still
| the underdog, but you profit over the long run by buying
| her at her too-low price. Those players are scooping that
| value wherever they can find it.
|
| For major electoral events that involve a populist with a
| large, loyal, and misinformed fanbase, the value pretty
| much finds you. Both of the described groups are making
| lots of money because the markets are wildly mispriced.
| jpcooper wrote:
| I wonder what you base your sentiment on. I can't imagine
| there's much juice left in Twitter sentiment analysis.
| chillydawg wrote:
| Heh.
|
| I used to work on those products, way back in 2008 or so. Let
| me save you some time: you won't get any action at good odds.
| Or, if you do, it'll be a tiny amount.
|
| Betfair run their own bots on those markets AND they get their
| bets into the market BEFORE the market opens to the public.
| Good luck anyway!
| jpcooper wrote:
| Thanks for the heads up. It's good to get confirmation that
| that actually happens. I'd be surprised if it didn't. I'm not
| too far from implementing my final idea to get profitable
| bets in. If that (probably as you say) fails, then it was a
| fun learning experience.
| plafl wrote:
| And they will cannibalize your profits, up to 60%, if you
| start to make significant money. And they can kick you out
| anytime they want. It's all in the terms and conditions.
| chillydawg wrote:
| IMO the premium charge is much better than commission as a
| pricing model. It's less predictable day to day but long
| term it better aligns incentives between the winning
| customers and the exchange. They don't really kick people
| off unless they're exploiting bugs in the exchange or
| engaged in illegal activity. They certainly won't ban you
| for winning on the exchange.
| aj7 wrote:
| Economic reasoning states that the vig will absolutely swamp
| any small edge you may discover.
| jpcooper wrote:
| Someone has to provide a market. That person has to do it
| profitably. I am assuming that this is possible, otherwise it
| would not be possible to play the game. My reasoning could of
| course be off. Could you elaborate?
| jonahx wrote:
| > Someone has to provide a market. That person has to do it
| profitably.
|
| But by this reasoning no one would play blackjack at the
| casino. That is, how do you know people laying these bets
| aren't just gambling at a disadvantage? You could view the
| whole thing as one more novelty game the casino is
| spreading.
| jpcooper wrote:
| I am assuming that all computer players of this game have
| access to the algorithm which I have discovered. I am
| assuming that all computer players have figured out the
| simple formula which gives them the cheapest profitable
| odds to back or lay an outcome given the commission and
| the outcome's probability (thanks to one of your previous
| comments).
|
| I am assuming that there will be punters who bite at the
| odds I provide. If there weren't, then the game would not
| be sustainable.
|
| I came across a "betting guide" for sale for this game
| which claims to advise on bets based on the "trends" seen
| in the game. I don't know what this means, but it gives
| me hope that there are people out there willing to part
| with their money in my direction.
|
| Failing that, the novelty was an interesting one, and
| it's all about the journey, dude!
| jonahx wrote:
| To clarify, I'm not saying your assumptions are
| necessarily wrong, or that you shouldn't try to test them
| and see what happens.
|
| I'm just proposing another possibility is that everyone
| playing on both sides is a "punter" having fun (or
| deceiving themselves). In any case, that everyone is
| losing long-term. With 5% vig I'd _guess_ that 's the
| most likely possibility.
| [deleted]
| rukuu001 wrote:
| These guys did it:
|
| _Walsh and Ranogajec took on the might of global gaming
| markets and managed to consistently win over three decades.
| Much like the most advanced hedge funds, which employ armies
| of PhDs to identify patterns in financial markets, Walsh and
| Ranogajec's consortium, called the Bank Roll, created
| statistical models that allowed them to exploit mispricings
| in betting odds wherever they could be found_
|
| https://www.afr.com/opinion/david-walsh-s-wisdom-beats-
| the-o...
| ximeng wrote:
| https://en.m.wikipedia.org/wiki/Vigorish For others who don't
| know what vig is
| kirykl wrote:
| Using a bot is acceptable under their terms?
| jpcooper wrote:
| Yes. They provide various APIs for this.
| ghgr wrote:
| Thanks for sharing (I recommend taking a look at his GitHub
| repo [2], where he nicely described the game and his approach).
|
| Some time ago I had a similar project with BetFair soccer
| matches. I assumed that the number of goals from each team is
| given by a Poisson process. Then, I used the largest market
| (Win/Lose/Draw) to estimate this parameter. With this
| information I could estimate the "fair" odds of more exotics
| bets (like e.g. team A will score 3 goals more than team B),
| which were often mispriced, and then use the Kelly criterion to
| estimate how much to bet.
|
| All in all a nice project, but... at the very end -just before
| deploying the bot- I realised that the BetFair Exchange is not
| open to Germans, so I wrote a blog post and open sourced the
| code. Still, a nice learning opportunity :-)
| jpcooper wrote:
| Thanks. I would like to point out again that the code is out
| of date, and the margins displayed there are miscalculated
| due to not accounting for commission. The general spirit
| remains.
|
| Did you backtest and show potential returns? Can you link to
| your model? Did you come across "Scoring dynamics across
| professional team sports: tempo, balance and predictability"
| [1]? They apply a Poisson model to team sports. I mentioned
| this paper to a company involved in betting who I interviewed
| with once, and they told me that they were doing something
| similar, but way more involved. It demotivated me somewhat
| regarding implementing it, making me wonder whether there
| really was any juice left to be squeezed from it.
|
| [1] https://link.springer.com/article/10.1140/epjds29
| ghgr wrote:
| I wasn't aware of that paper, but it seems we came to a
| similar conclusion. From the paper:
|
| > Across all sports, scoring tempo - when scoring events
| occur - is remarkably well-described by a Poisson process,
| in which scoring events occur independently with a sport-
| specific rate at each second on the game clock.
|
| So, yes, basically that was my concept. Estimate its
| parameters and then you'll have the "real" odds. Any
| difference with the market ones is a potential of profit
| (considering the house fees).
|
| I couldn't backtest it since I couldn't find free
| historical data for all types of markets (then again, maybe
| I didn't look hard enough). I forward tested it, though,
| and it seemed to work.
|
| My code is here [1], which also links to a blog post
| describing the method. Feel free to take a look and contact
| me to discuss these ideas.
|
| Also, if you're interested in mathematically modelling
| probabilities, we are looking for team members to develop a
| product to estimate the risk of default in loans. After
| all, a loan is another kind of bet, that takes place in a
| slightly harder to model environment.
|
| [1] https://github.com/ghgr/BetFair_Arbitrer
| jpcooper wrote:
| Thanks for the tips. The loans idea sounds interesting. I
| will send you an email.
| throwaway98797 wrote:
| Any of his books worth reading?
|
| Group Representations In Probability And Statistics
|
| Magical Mathematics: The Mathematical Ideas That Animate Great
| Magic Tricks -- winner of the 2013 Euler Book Prize
| zests wrote:
| Persi is always worth reading although I have only read his
| papers.
| sdenton4 wrote:
| "Group Reps" is one of my favorite math books of all time. It
| is a fantastic read, and completely changed the way I think
| about representation theory. (In short, you can generalized the
| Fourier transform to functions in groups and take advantage of
| all the usual Fourier tricks and intuitions.)
| jaytaylor wrote:
| Link to pdf: http://personalpages.to.infn.it/~billo/didatt/gr
| uppi/reps.pd...
|
| Caveat emptor: Not sure if it's the right book, though. No
| author listed.
| iainctduncan wrote:
| Magical Mathematics is great! Dense actually, walks the
| "layperson" through proving how some complex card magic works
| with combinatorics. Really fun read, some mind bending stuff in
| there! The math is harder than one would expect the cover...
| keepshoutingsir wrote:
| Tl;dr: There's no particular bias to heads or tails, but a coin
| will land the same way it was when it was tossed 51% of the time.
| sp332 wrote:
| I remembered the result that showed a coin toss was biased that
| way, but I didn't know that the degree of bias hadn't been
| nailed down. I guess it makes sense that it varies from person
| to person. Now I'm interested in an anthropological study of
| how different people toss coins...
| eh78ssxv2f wrote:
| Do we know the reasoning behind the bias?
| hnracer wrote:
| I'm probably being dense but what do you mean by "the same way
| it was when it was tossed 51% of the time"?
| kilbuz wrote:
| When you flip a coin, it is either heads side up or tails
| side up as it rests on your thumb. This initial position may
| slightly affect how it ultimately lands.
| [deleted]
| callamdelaney wrote:
| Pretty sure it's because the heads is heavier.
| Closi wrote:
| It's not, it's because when flipped the number of flips the
| coin does in the air is not entirely random.
| morelisp wrote:
| Not only is this (somewhat obviously given the discovered
| bias) wrong, it's physically impossible to bias a coin flip
| by weighting one side. Weighting a coin will change its
| center of gravity but to bias a flip you would need to impart
| angular momentum _while it is in the air_ ; at any constant
| rate of spin (again somewhat obviously once pointed out) each
| side will spend 50% of its time "up" and 50% "down".
|
| https://www.tandfonline.com/doi/abs/10.1198/000313002605
| callamdelaney wrote:
| Why does toast land butter side down so often?
| morelisp wrote:
| If you flip toast rather than drop it, it doesn't. Also,
| if you drop it it doesn't really land "butter side down"
| but "top side down". Just butter the bottom of your toast
| to solve the problem!
|
| https://en.m.wikipedia.org/wiki/Buttered_toast_phenomenon
| ppezaris wrote:
| There's a huge assumption in this assertion: that you catch
| the coin.
|
| If you let it bounce on a table, it's much more likely to
| land heavy-side down. Just imagine a very thick coin one
| side lead the other side balsa wood. That would land lead-
| side down much more than 50/50 if allowed to bounce on the
| table.
| 0-_-0 wrote:
| I thought it was common knowledge that heads are heavier than
| tails. My head certainly is.
| Spooky23 wrote:
| Some folks have big heads, others big tails.
| prionassembly wrote:
| HN is way too harsh on jokes, even when they're really good
| - like this one.
| rootusrootus wrote:
| I think good jokes can get appreciation here, but we try
| pretty hard to avoid the reddit-style puns and joke
| chains that distract from the discussion. They're
| entertaining sometimes... on reddit.
| umanwizard wrote:
| There are plenty of other sites you can go to if you want
| to read chains of joke comments.
| keithwhor wrote:
| The article is more interesting than just the finding, to those
| who want to delve deeper. It's a short biography of Persi
| Diaconis, an accomplished mathematician with no high school
| diploma who didn't start studying mathematics until he was 24.
|
| Really, really fascinating life and experience.
| dheera wrote:
| I think it's more random for me because I flip coins onto the
| floor, not onto my hand, or spin them on the table and see how
| they land when they stop spinning.
| thoughtstheseus wrote:
| With practice most people can dramatically influence the outcome
| of a coin flip. I've gotten to >80% accurate when I was
| practicing.
| throwaway2245 wrote:
| In this sense, I found the 51% surprising (surprisingly close
| to 50%).
|
| If you have a very practiced, consistent flip, you can always
| get the same result.
| jcheng wrote:
| The article notes that he got to 100% when intentionally
| biasing the flip, the 51% was for ordinary people doing
| ordinary coin flips (no attempt to bias).
| throwaway2245 wrote:
| To clarify, this is exactly what I found surprising: that
| any unintentional bias that exists, in those ordinary
| cases, cancels out almost (but not quite?) perfectly.
| unstatusthequo wrote:
| Appreciate the much-needed TLDR on this.
| spiffytech wrote:
| What do you do to bias the outcome?
| [deleted]
| nojokes wrote:
| You catch it at the right moment. You learn to feel the right
| timing by practice.
| [deleted]
| david927 wrote:
| Flip it with the right amount of force, like a juggler with
| bowling pins.
| alisonkisk wrote:
| No. You need to put a lot of horizontal torque on it so it
| never flips over at all.
| thoughtstheseus wrote:
| That's not a coin flip! I agree that's a method to
| achieve the outcome though. If you get the wobble right
| it's almost imperceptible.
| iainctduncan wrote:
| no audience is going to fall for that...
| thoughtstheseus wrote:
| They do, occasionally it doesn't wobble enough and looks
| obvious but done right it's not perceptible. As a tip do
| it physically close to someone and catch it high so they
| have a worse angle to see the lack of flipping. For a
| real "performance" there is significantly better methods
| though.
| namenotrequired wrote:
| From the article:
|
| > Diaconis himself has trained his thumb to flip a coin and
| make it come up heads 10 out of 10 times
| TylerE wrote:
| 10 out of 10 isn't really that impressive.
|
| If his actual success rate is, say, 70 out 100 10/10 is easy
| enough to get by chance.
|
| 990 out of 1000 would be much more pressive.
| moonchild wrote:
| It's a popular piece. '10 out of 10' probably just means
| 'reliably'.
| thoughtstheseus wrote:
| Flip the coin the same number of times, catch consistently and
| get good at adjusting the catch. Use a large and heavy coin
| like a silver dollar. - former magic enthusiast.
| tossthere wrote:
| Former Pokemon TCG enthusiast (circa 1999) here! Kids back
| then had the same approach, large and heavy coin, flip from a
| consistent height, consistent strength. Heads every time.
|
| There were popular coins made specifically for Pokemon TCG
| then, and they happened to be large and thick and plastic and
| heavy, so the cheating was widespread.
|
| I was one of the many kids who did this routinely. But there
| was one kid who took first place nearly every week, out of a
| field of 50+ entrants. "He must cheat the coin, AND be an
| amazing deck builder and card player on top of that", I
| thought.
|
| Then one week I was matched against him, winner goes to top
| 8.
|
| Our decks were nearly identical, I could flip heads every
| time, I was a strong player. "50-50" I thought.
|
| Turn one. Retreat Scyther, in Electabuzz, Thundershock. I
| flip my giant coin, and before it flattens, he picks it up
| and hands it back to me. "Flip it higher please."
|
| He must know I'm cheating. I assumed he must be cheating too
| though, every other kid is, and he wins this tournament every
| week!
|
| Guilty, I flipped the coin fair this time, higher. Before it
| flattens he picks it up again, hands it back to me. "Higher
| please."
|
| I lost every coin flip that match. He won, went to the top 8,
| later won the tournament for the nth week in a row.
|
| It wasn't until adulthood that I realized how he was winning
| every tournament. In a field full of cheaters trying to flip
| heads every time, all he had to do was wait for the coin to
| enter that terminal spinning motion every coin does right
| before it flattens. Well practiced at this, if he sees that
| it's about to flatten heads, he quickly grabs it and hands it
| back to you. "Higher please." Repeat until the coin flattens
| tails.
|
| His tactic was a silver bullet in a field full of flip
| cheaters. Nobody ever questioned his intentions when he
| grabbed the coin and asked for a higher flip.
| mundo wrote:
| I can imagine that on a slow, lazy flip with a big coin, but
| is this realistic with what I think of as a "normal" flip,
| i.e. a US quarter, flipped rapidly end-over-end with the
| thumbnail, 2+ feet in the air? It's hard to believe that even
| decades of practice would get this reliable, and I don't
| think I've ever seen a stage magician rely on it in a
| routine.
|
| For comparison, the other method (where the coin rotates
| about its vertical axis) is something you can learn to do
| tolerably well in a few minutes, and to do very well after a
| month or two at most. The motion of the coin will look
| perfect, and you can even get the "ding!" noise of a real
| flip. The only giveaway is the hand position; a real flip is
| done with the thumb under the coin, and this trick (for me,
| anyway) requires the thumb to be on top.
|
| Source: taught myself to do this many years ago. I'm not a
| magician and have never used it to win money, I just use this
| to settle arguments between my kids.
| murkle wrote:
| > Preliminary analysis of the video-taped tosses suggests that a
| coin will land the same way it started about 51 percent of the
| time.
|
| From 25 trials?
| morelisp wrote:
| They're measuring rotational velocity, not outcomes. As
| described in the article, actually flipping to determine the
| bias would have been a) arduous, b) probably prone to a margin
| of counting error larger than the possible bias (as they
| discovered for dice).
| forgotmypw17 wrote:
| I'm very surprised that a scientist who previously worked with
| shaved dice did not take into account or even mentioned what kind
| of coin they were working with. Many coins are not even close to
| equal weight or shape on each side.
| hardmath123 wrote:
| See also: this short piece about detecting bias in coin tosses by
| listening to the "ping" they make when tossed.
| https://cs.stanford.edu/~kach/can-one-hear-the-fate-of-a-coi...
| thotsBgone wrote:
| I did a high school math project that was like this but with
| dice. After thousands of die rolls, I found out that dice with
| pips are not very fair, but dice with engraved numbers are
| better.
| FpUser wrote:
| >"Diaconis didn't even own a computer. (He still doesn't. He says
| he got sick of adjusting to new operating systems and noted,
| after not using computers for several years, that "nothing bad
| happened to me. And I said, 'Well, I could either learn the
| current system or learn differential geometry. I think I'll learn
| differential geometry this year.'")"
|
| Good for him. He made the right choice I think.
| crazypython wrote:
| Sounds like he would enjoy GNU Emacs, which hasn't changed its
| interface in decades. Even the copy-paste shortcuts are the
| same.
| simonh wrote:
| The same is true of Notepad. It's not really the point.
| crazypython wrote:
| Emacs has calendar, weather, email, games, todo, search,
| and almost everything else you need in a modern operating
| system.
| _jal wrote:
| Having spent the majority of my days typing at various
| applications for the last 30 years, I have developed a great
| appreciation for UI longevity.
|
| I'm a vi person by muscle-memory, and the stability there
| means I've learned a thing or two about it. But I've used
| emacs enough that I'm comfortable there, too, at least to a
| point.
|
| Don't get me wrong, I love playing with new toys. But I also
| love not having to think about my tools - I'd rather think
| about what I'm building.
| aj7 wrote:
| I'd rather give myself an enema. Downvote me.
| PragmaticPulp wrote:
| The key to breaking this potential advantage is to let someone
| other than the coin flipper call heads or tails while the coin is
| in the air.
|
| That's how I was taught to do it as a kid when we were playing
| for fun. I never really thought about the reasoning until now.
|
| If you talk to people deep into the performance magic community,
| there are myths and legends of people who practice dice rolling
| to the point where they can influence the outcome of a dice roll.
| I suppose training for hours a day for years on end could have
| some such influence.
| pschw wrote:
| It is definitely doable. _Scarne on Dice_ (1945) details a
| number of methods for controlling dice throws. And Steve
| Forte's "Gambling Protection Series" videos show what the moves
| look like in action.
|
| Mind you, none of these moves will fly in a modern casino.
| wool_gather wrote:
| Also the book "Gambling Scams" by Darwin Ortiz.
|
| As I recall, the techniques generally involve _not_ making
| normal "fair" free-tumbling rolls, but constraining the die
| movements somehow. Things like tossing them into a corner or
| making one of the pair spin flat, around only a vertical
| axis.
| daveFNbuck wrote:
| That doesn't break the advantage, it just mitigates it. The
| advantage is just small enough that it doesn't matter much even
| before the mitigation. The caller could still look at the coin
| before it's flipped and have a slight edge in calling it.
|
| It's not terribly hard to influence the outcome of a die roll.
| You can roll it end over end like a wheel so two of the faces
| remain on the sides and can't be the result.
| tzs wrote:
| There was an episode of the old "Breaking Vegas" series [1] on
| the History Channel about such "dice dominators". This site
| seems to have the episode online [2].
|
| [1] https://en.wikipedia.org/wiki/Breaking_Vegas
|
| [2] https://www.goldentouchcraps.com/HistoryChannel.shtml
| donpdonp wrote:
| The method to get fair flips from an unfair coin is interesting
| and was recently on hackernews. say a coin is heads(H) 70% of the
| time. that means it must be tails(T) 30% of the time. If you flip
| it twice, and calculate the odds of each pair of flips (multiply
| the probabilities) - HH 49%, HT 21% TH 21% TT 9%. The middle two
| outcomes are equal probability, aka a fair coin. So ignore HH and
| TT flips and use HT and TH as Head and Tail of the 'fair' coin.
| btilly wrote:
| The Von Neumann method depends on coins being independent.
|
| But the measured bias here is a 51% chance of getting what you
| started with. Which means that if you flip over and over again,
| there is a correlation between consecutive entries.
| dragontamer wrote:
| Just note: you're throwing away 58% of your results! The
| original coin flip runs 230% the speed of your new RNG (that
| throws away so many results).
|
| The Von Neuman method is a great way at demonstrating that
| removal of bias is possible, but not necessarily practical
| (especially at high bandwidth). In practice, you want to just
| cryptographic_hash(coin flips), which should maximize your
| entropy up to the entropy limit of 1/2 the hashsize (assuming
| you have a perfect cryptographic hash function).
|
| A 512-bit perfect cryptographic hash can only support 256-bits
| of entropy, due to the birthday attack.
| typicalset wrote:
| In the general case, you can compress a sequence of N biased
| coin flips with arithmetic coding. For a coin with known
| bias, this compression is (essentially) optimal and will
| therefore produce ~N*(Shannon information) unbiased bits.
| dragontamer wrote:
| In the "even more general case", the Shannon Theorem
| suggests that random_code(message) -> reaches the Shannon
| Limit. :-)
|
| The problem is that perfectly-random codes can't be decoded
| very easily... so this little factoid is kind of worthless
| in most cases. Fortunately, we don't actually care what the
| information looks like when flipping coins. We usually just
| want "some random message", so no need to decode in this
| application!
|
| ------
|
| Cryptographers assume hash-functions are perfect random
| codes (they aren't in theory, but they are in practice...
| at least until they're cracked. Kinda funny how that works
| out). As such, the cryptohash(message) methodology should
| also reach the Shannon-limit, as long as your cryptohash
| remains secure... and you stay within the blocksize
| restrictions of the hash.
| MrQuincle wrote:
| What about just after the unfair coin sequence toggling every
| even position?
|
| HTHHTHHTTHHH
|
| HHHTTTHHTTHT
|
| Should be fair. Though there is order now when the coin was
| unfair.
| jonahx wrote:
| Related: https://fivethirtyeight.com/features/can-you-make-an-
| unfair-...
| nobrains wrote:
| Slightly related, and of interest to minecraft players and speed
| runners: https://www.youtube.com/watch?v=1iqpSrNVjYQ
| lemonspat wrote:
| > Diaconis himself has trained his thumb to flip a coin and make
| it come up heads 10 out of 10 times.
|
| I had better dexterity as a child, and after hours I was able to
| flip a coin to 'heads' about 8/10 of the time. Same position,
| same muscle reflex, same results. I would proudly show my family
| my skill
|
| Over time, I lost interest and the skill
| [deleted]
| iainctduncan wrote:
| Perci is a fascinating and very accomplished guy. His book
| "Magical Mathematics" is really interesting. There's actually a
| long history of math and comp sci folks who were very good
| magicians. Alex Elmsley, Bob Hummer, Martin Gardner are a few
| other to check out if you're into that stuff. (says the guy who
| has spent the last few days alternating between hacking and
| vainly trying to master the perfect Faro shuffle... ).
| emcq wrote:
| Don't forget Ron Graham who cowrote magical mathematics!
| iainctduncan wrote:
| Yeah! And Ron was a pro juggler too. (as have been I!) Sadly
| Ron died this year.
| KwisatzHaderack wrote:
| Years ago, I went to a talk Persi Diaconis gave on this exact
| topic. He was the most charismatic mathematician I ever met in
| person.
| kazinator wrote:
| Half-assed tosses of a flat object can obviously be biased. We
| can predictably flip an object to the other side with a light
| toss, not to mention flip a pancake.
|
| If you flip a coin through the air so that it rapidly rotates
| (say more than 15-20 times) before being caught, that seems like
| it would erase the bias.
|
| If there isn't decent spin, then "all bets are off" (quite
| literally).
|
| As a matter of procedure, there should be a rule: the coin tosser
| must place the coin onto index finger, which is coiled around the
| thumb, such that just the edge of the coin is struck from below
| by the thumbnail, when the thumb is flicked, sending the coin
| upward in rapid rotation. There should be additional rules that
| the coin should attain a height of at least one head height above
| the top of the tosser's head, or something of the sort.
| 10000truths wrote:
| Better yet, just bounce the coin off of two surfaces before it
| lands. Makes it almost completely unpredictable.
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