40k coin tosses yield ambiguous evidence for dynamical bias | Hacker News Reader

40k coin tosses yield ambiguous evidence for dynamical bias

stat.berkeley.edu

187 pointsgeocrasher117 comments

> A first comment is that it would have been better for each individual to have done both "Heads up"and "Tails up" tosses (which was part of the intended protocol, but on this aspect of the protocol there was a miscommunication)

That was a pretty unfortunate error in the experiment. Maybe it doesn't matter but now we don't know. It would also have been nice to have them swap coins halfway to expose any individual per coin biasing. It may seem like an irrelevant thing but I've been a lifelong magician specializing in advanced slight of hand coin magic. I carry a set of coins with me all day, every day and handle them constantly. It started out as practice but evolved into both practice and a kind of fidget toy. I have sets of coins I've probably handled for thousands of hours over decades.

Most people think of coins as immutable but they actually change quite a bit after hundreds of hours of handling. Most advanced coin magicians don't tend to use "trick" coins from a magic shop because they are actually too limiting. The coins I use are completely normal circulated coins but they are very specific because there are subtle differences in how coins handle which, at the most advanced levels, can matter. I have year-matched sets of coins I've carefully assembled because they have the degree of surface wear (sometimes called 'softness') and edge-milling which works best for the style of slights I do. Coins also vary in shape and many aren't quite round. I've actually hired a specialized machinist (aka coin-smith) to 'true-up' the shape and then re-mill the edges of certain coins.

Based on my admittedly unusual experience in handling coins, I suspect that weight, edge and surface variations in individual coins could have a material aerodynamic impact at this statistical level (sub-half a percent). BTW, there are coin magicians who have mastered the ability to flip a normal coin and control the outcome to >95%. While the coin is normal in every way, it does need to be a coin they've specifically trained with. Otherwise the hit rate falls considerably.

Unrelated question: how do your hands look after handling coins so constantly as you do? Is there a slight sheen to them? Do you have less hair on your hands than you normally would, or more?

No, there's no noticeable difference in the skin surface of my hands. Over years of practice, serious coin workers do develop substantial hand muscle strength and joint flexibility but that's not visually noticeable. It's also just a byproduct of practice and not needed for most of what we do. The key physical ability isn't applying force, it's actually enhanced sensitivity. I can feel precisely where the coins are and feel how the weight is shifting as they move. This sensitivity enables precise control which is really the key thing.

The same is true for card magicians, and card handling. I have a lot of things I can do with my hands and joints that aren't something you can see.

Same with enhanced sensitivity. I can fairly accurately tell you how many cards I pick up from a group. It's just practice, once you can do 2, you do 3, all the way up to 10, to 15 to 20.

Is that a kind of subdigitizing?

"Subitizing" ?

Funnily enough, but also not surprising, it's the same for drummers and their preferred type of sticks.

It is called "kinesthesia", which is important in playing musical instruments, sports, flying aircraft, etc. With respect to flying, pilots trained by Army (navy, airforce) develop these skills far better than those directly recruited by carriers from colleges. We can see those results in some fatal crashes (Air France 447, Asiana 214). I am not blaming the pilots. When one flies with the aid of machines (fly by wire, simulators), one doesn't develop the kinesthesia required to get out of tricky situations(when instruments don't work or when instruments misread, etc).

Imagine being in the room when they realized that, after forty thousand coin tosses, they screwed up the experiment. Oops.

Then it becomes an 80k coin toss experiment :)

This is how ML training goes. My longest run was 2 months before it collapsed.

But did you learn anything?

Results of analysis:

Well it's not like we're gonna run out of undergraduates, just do the experiment again.

That's super interesting! Of note, the book Quicksilver (First book in the Baroque Cycle) goes into a few diatribes about subtle differences in coins in Europe a few hundred years ago; eg coins being valued by metrics other than their face value.

Eg: > “I say, Daniel, is it true what they say, that those coins are perfectly circular?”

> “They are, Isaac—not like the good old English hammered coins that you and I carry in such abundance in our pockets and purses.” ... >“if someone clips or files a bit of metal off the edge of a round coin with a milled edge, it is immediately obvious.”

> “That must be why everyone is melting those new coins down as fast as they are minted, and shipping the metal to the Orient…?” Daniel began,

> “…making it impossible for the likes of me and my friend to obtain them,” Isaac finished.

> “Now there is a good idea—if you can show me coins of a bright silver color—not that black stuff—I’ll weigh them and accept them as bullion.”

> “Bullion! Sir!”

> “Yes.”

> “I have heard that this is the practice in China,” Isaac said sagely. “But here in England, a shilling is a shilling.”

> “No matter how little it weighs!?”

> “Yes. In principle, yes.”

> “So when a lump of metal is coined in the Mint, it takes on a magical power of shillingness, and even after it has been filed and clipped and worn down to a mere featureless nodule, it is still worth a full shilling?”

> You exaggerate,” Daniel said. “I have here a fine Queen Elizabeth shilling, for example—which I carry around, mind you, as a souvenir of Gloriana’s reign, since it is far too fine a specimen to actually spend. But as you can see, it is just as bright and shiny as the day it was minted—”

> “Especially where it’s recently been clipped there along the side,” the lens-grinder said.

> “Normal, pleasing irregularity of the hand-hammered currency, nothing more.”

Are we all in the same book club or something?

A substantial chunk of the books I read are based on HN recommendations, so... yes?

Egghead book club. Neal Stephenson, Greg Bear, Greg Egan…

Go on...

graeber

If you like that book, read the Newton biography "Never at Rest."

Almost certainly Stephenson read it in his research, and he gets the nature of Newton totally correct in fictional form.

Bought!

> BTW, there are coin magicians who have mastered the ability to flip a normal coin and control the outcome to >95%.

I managed 11 heads in a row with a quarter when I was in high school. I was intentionally going for heads so it was either beginner’s luck or an absurd statistical fluke.

If you're not using a specific technique, that was just a fluke.

I'm not a magician, I just learned this one trick/set-of-tricks (and I'm not 95% success at it either).

Even when not trained if you are basically looking to "repeat" a motion, you can get a reasonably consistent flip. A lot of folks think these far out results are just statistics, but statistics actually tells us how incredibly rare this would be.

11 in a row is perhaps still part fluke without training, but as long as you were trying to do the same thing again not unreasonable.

Please send us further down this rabbit hole, it sounds fascinating!

What sort of flourishes/fidgets do you find yourself gravitating to most?

What does great sleight of hand coin magic look like? Who do you admire most?

Where should someone get started if they want to explore this?

> What sort of flourishes/fidgets do you find yourself gravitating to most?

There are a huge variety and it's mostly down to personal preference. Popular flourishes include coin rolls and coin stars. Popular slights include dozens of different palms with single and then multiple coins.

> What does great sleight of hand coin magic look like?

Done right it can be absolutely mind-blowing. For an example take a look at some of Danny Goldsmith's videos (https://www.dannygoldsmithmagic.com/). Danny is very, very good but like a lot of specialized skills, those who aren't deeply into it won't be able to notice a meaningful difference between the top 20% of coin workers.

> Who do you admire most?

That's really down to personal taste and style. One unusual thing about magic is that the "best magicians" in the eyes of other experienced magicians are generally people you've never heard of. Fame doesn't really correlate with the pinnacle of skill. While most famous magicians like a David Copperfield are skilled, they would be the first to tell you they can't hold a candle to the most skilled coin workers or card mechanics.

> Where should someone get started if they want to explore this?

For serious coin work, Danny's teaching videos would be a good start. Look for ones he flags as being appropriate for novices. For learning the art of magic in general, I'd suggest not wasting money buying individual "tricks" unless you just want a few easy party tricks to amaze (or annoy) friends with. Most people who get deep into magic discover that the most valuable and broadly applicable knowledge comes from books, videos and live learning (called "sessioning" by close-up workers). I don't buy much magic in recent years because I'm at the point of just perfecting skills, so it's hard to recommend an online store but a safe bet for beginners would be Vanishing Inc. https://www.vanishingincmagic.com/. The guys that run VI are deeply experienced and they seem to avoid selling a lot of the 'crap magic' that looks amazing to novices but isn't actually all that useful. I was at a lecture by the co-founder a few weeks ago and the guy not only has deep knowledge and mad skills, he clearly loves magic and is good at teaching.

If you just want to watch some higher-quality magical performances, I'd suggest Penn & Teller's "Fool Us" show (lots of clips on YouTube). The technical coordinator on that show, Michael Weber, is a long-time magical inventor and author who is well-regarded by other magicians. He and Teller work together to curate the acts that get on the show and you're basically getting to see some top notch talent hand selected by guys who know the difference.

> BTW, there are coin magicians who have mastered the ability to flip a normal coin and control the outcome to >95%.

Now you made[1] me want to rewatch Rosencrantz & Guildenstern Are Dead[2] again...

[1]: https://www.youtube.com/watch?v=C_TfdNAXOwE

[2]: https://www.imdb.com/title/tt0100519/

I missed it when it came out. Dream casting.

> BTW, there are coin magicians who have mastered the ability to flip a normal coin and control the outcome to >95%

I guess it depends on coins, but my experience with French coins is that it's extremely easy to get stable outcomes. Without training or anything, if I start with heads, I almost always get heads, and if I start tails, I almost always get tails. I could totally believe that with training, I could pick the result. But if you let me pick the side that is up when tossing, then I can predict the outcome.

I used to play a MUD (a text-based proto-MMORPG). It had a command to let you flip a coin. After many years, a player felt like they were noticing more tails than heads. They ran some experiments, and eventually an administrator checked the code - it turned out there was an off-by-one error in the coin flip logic (which inexplicably relied on generating a random number 1-10), such that coin flips had been 40/60 since the release of the game, and no one had noticed for over a decade.

30/60 maybe?

EDIT: probably not after reading the comments. I thought I was smart ;)

Either could be the case - if it was generating numbers from 1-9 instead of 1-10 30/60, if it was comparing <= 5 vs <5 it could be 40/60

Yeah, I'm almost certain it was the latter. I remember they shared the code snippet, but that was in 2007 and the game's forums have been replaced since then. I did find a recounting of it:

https://forums.achaea.com/discussion/comment/249213/#Comment...

(and the associated quest to discover the easter egg of a coin landing on its edge with 1 in a million odds that was added when the bug was fixed)

40/50 probably. One of the 10 numbers never showed up. 1-5 tails, 6-9 (should be 10) heads.

I ran into the same thing when I made a custom dice roller for Settlers of Catan. Rand(1, 6) never produced a 6, which you could tell if you inspected the comments closely, but still feels counter-intuitive to me.

Our Catan set has noticeably nonuniform dice. They roll high numbers much more frequently than low numbers; presumably bc the high-numbered sides weigh less (?). We haven't buckled down and done the science but it definitely tilts the game play, to the extent that we're considering swapping them out for new dice.

(or this could just be Catan Crankiness (TM), but we've all noticed it quite a bit...)

On the subject of Catan and dice:

https://www.chiark.greenend.org.uk/~sgtatham/dice/

That is a good example of an over engineered solution. The game already has a far simpler way to make it easier for all players to get started, regardless of how many turns you play. It might be from one of the expansions, but we use it in the normal game as well.

The "friendly robber" rule states that you are not allowed to place the robber on any tile that has a village of a player that does not have any points yet.

Which MUD was this?

Achaea

What are the odds wow

Huh. I once wrote a MUD client that didn't have an obviously documented way to quit and got email for years after complaining that I had "made the application too addictive".

My favorite baffling RNG bias was in the game Maplestory. It always seemed like RNG was extremely "clumpy". For example, critical hits would frequently come in bursts - you might get around 5 seconds of non-stop crits.

The crafting system was heavily chance based. You'd get say 7 attempts to augment a piece of gear, using 'scrolls' that had a certain percentage chance of succeeding. The lower the chance, the bigger the improvement to your gear. So a pair of gloves that had 7 successfully applied +attack 10% scrolls would be incredibly valuable. The superstitious method to crafting (which, anecdotally, worked incredibly well) was to get a ton of gloves and a ton of scrolls, and apply one scroll to one pair of gloves and throw them away if it failed. Once you had a single success, you would apply the scrolls as quickly as possible to try and ride the RNG wave. In my experience, this would very frequently result in getting 3+ successes in a row.

Many games implemented strange RNG mechanics like this to game psychology. Eastern 2D RPGs were notorious for this, since they relied heavily on MTX.

I'd love to see the code!

A funny story of a miscommunication about randomness in the real world, there was a miniatures game called Heroclix in the early 2000s. They apparently communicated with the factory that was making them in China that they wanted 5% of the miniatures printed to be the ultra rare ones.

So the Chinese happily obliged. The first 5% of the run was nothing but ultra rare miniatures and none of the remaining 95% printed had any ultra rates. The first people to buy boxes were very happy!

Discworld MUD, if you flip a coin in a room where to many spells have been cast it might turn into a caterpillar.

From a physics stand point, coin tossing is similar to knife throwing or axe throwing. It's completely within human capability to intentionally or unintentionally time the toss with some degree of accuracy. I doubt anyone can be "good" at this (otherwise it would be a great grifting trick), but surely this creates at least a marginal bias in the data at a high enough scale.

Even that poses a fun test. What flipping method is needed to ensure some level of randomness.

I love that idea of a grifter being able to flip a coin as they choose.

Probably a device that would uniformly toss the coin with regards to force applied at X number of different strengths.

Or have both people flip a coin secretly, then reveal the coins and XOR the results :D

Maybe letting the coin fall on a hard surface, so that it has a chance to bounce/spin randomly before falling flat (or staying on its side!)

When I was around 15 I tried to master the controlled coin flip, having been exposed to the idea in one of the Stainless Steel Rat books by Harry Harrison. I reached around 80% success flipping for heads, so long as I used the catch-and-show method. It was easier with heavier and bigger coins, tougher with lighter and smaller ones. Letting the coin drop the the ground and bounce took me back to 50%.

Satisfied that it could be done, I moved on - never used the skill, except as a party trick.

This is why the casino people get upset when you roll the dice at a craps table and continually avoid hitting the back wall. The rubber pyramids lining the wall are critical to ensuring they maintain their edge :)

Same here. Long ago I tried to generate an encryption key by flipping a coin. After getting a rhythm and flipping heads 20 consecutive times I gave up.

You could still XOR longer runs together, reducing the bitrate but mostly eliminating the bias.

There is a protocol for this... you flip the coin twice. HH or TT you ignore, HT or TH you keep (and treat as H or T). This eliminates any bias towards H or T that the coin or flipping procedure may have.

Antidotally I can confirm that it is possible toss a coin like that. It helps have a larger coin and to impart it with relatively little rotation.

I can consistently flip a US quarter and get the desired outcome. I have not measured accuracy but is definitely 80% or more. This can be with a quarter I just picked up at a store… no special wear.

From what I've heard, Perci Diaconis (one of the authors of the original paper) actually could do this. He was a magician before he became a mathematician, and a lot of his early mathematics work focused on math relating to the magic tricks he used to do

Art Benjamin at Harvey Mudd College is another magician/mathematician. The two skills line up pretty well, it seems. When I got my math teaching credential there were two people in my cohort who did magic tricks as well as at least one teacher at the school where I did my student teaching.

In roulette this is known as the "dealers signature" where a bored zoned out dealer can sometimes hit the same region/sector of the wheel on consecutive spins.

I've seen this happen (or at least appear to happen) in real life where an obviously bored dealer was consistently hitting the same 1/3'd of the wheel and players were taking advantage of it. After a while a suit shows up and starts giving heat not to the players but to the dealer. Chatting them up with nonsense conversation to distract them out of their zone. This wasn't a pit boss/supervisor but casino security - guys that emerge from back rooms to give heat to card counters in blackjack.

The distraction worked: the now very awake and nervous dealer was no longer hitting similar areas and the players moved on to other tables.

People can definitely get "good" at coin tossing. It isn't even too difficult a trick to master. It's not really possible to use it in any kind of grift scenario though, because there aren't any real world cases where you can win money (or gain any advantage) just by tossing a coin a certain way. Rolling dice has a lot more potential, but those are in turn heavily monitored, e.g. in casino settings.

> there aren't any real world cases where you can win money (or gain any advantage) just by tossing a coin a certain way.

Paying off a referee to have the coin turn up heads in overtime in a football game seems like it may net you some profit.

If you can pay off a referee then the coin toss would be the least effective way to do it. A single bad call would gain you a much bigger advantage.

It has far more plausible deniability, however.

Maybe. It only takes one bad call at the right time to throw a game. If the ref is consistently bad and missing call for both sides what is one more bad call?

I don't care much about bad refs so long as they are consistently bad. However a perfect ref who missing one critical call is a lot more suspicious. Of course I prefer a great ref, but I can work with a fairly bad ref.

I believe NFL coin tosses let the coin hit the ground. All methods I know of biasing a coin toss involve catching the coin. As long as the coin hits the ground and bounces at least once, a lot of randomness is re-introduced.

Forcing a coin toss of the sort described in the article is well within reach of the interested stage magician.

Forcing a coin toss of the kind used in sport, where the coin hits the ground, is more difficult.

You don’t have to be a magician. I was impressing other kids doing exactly this, you just toss so the coin goes high but turns slowly and catch it in your hand rather than let if fall to the ground. After not so many tries you get a pretty good intuition on how you have to throw to get the desired result.

It works the other way around, this kind of trick is (stage) magic so if you're doing it, hey presto, you're a magician.

There's also a particular method to get the coin in a tilted spin that looks (and sounds) _remarkably_ like a real flip, and is I think going to be more repeatable than what you're describing.

It wouldn't be. People who fall for shell games have to accept that it's fishy and just want to show they are better.

If the coin were unbiased, we could compute the exact probability of getting 10231 or more heads with 20000 flips as:

"sum (20000 choose x)/2^20000 for x from 10231 to 20000",

which Wolfram Alpha evaluates to 0.00056.

The probability of getting a number of flips that differs from 10000 by at least 231 is twice that, so about 0.001.

So, in fact, the probability of this happening by dumb luck is about 1/1000. That's pretty strong evidence.

They mentioned that when A tossed the coin, B would record the result on a spreadsheet. But did A know the result of each coin toss as it happened and read it off to B, or did B whisk the coin out of A's hand without A's knowledge of the outcome? Ideally it should be without knowledge of the outcome, so that A's tossing style wouldn't be subconsciously influenced by the result in a kind of self-inflicted Clever Han(d)s effect.

I think that every physical system is a little biased. That's what they change the balls at lottery for every game.

I've read an article about a guy who was observing frequencies at roulette long time ago and, based on that, he made some wins. The casino learned that and switched the tables each day, so he wasn't able to win any more.

There's someone who built a craps table and perfected exactly how to throw the dice. If you control the inputs, you control the output.

I could see someone learning exactly how to flip a coin to control how it lands, or at least greatly influencing the outcome.

Discussed here: https://news.ycombinator.com/item?id=30489022

I think my approach would be to build a coin-flip machine (well, several) that could operate independently, then use computer vision to get the final readout (the result of the flip). Then, if it deviates from expected, use a high speed camera to watch the coins. Oh, and randomize everything about the trials.

Take a look at "Dice-O-Matic hopper and elevator" http://gamesbyemail.com/News/DiceOMatic HN discussions https://news.ycombinator.com/item?id=14806986 (246 points | July 19, 2017 | 57 comments) and https://news.ycombinator.com/item?id=626092 (81 points | May 26, 2009 | 4 comments)

yes, that's a design. It's hard to see the details because the video is poor and the explanation is a bit rambling. It looks much more complicated than necessary and from what I can tell, the rolls themselves aren't truly independent because dice can interact with each other.

As in the paper referenced in the OA?

https://statweb.stanford.edu/~cgates/PERSI/papers/dyn_coin_0...

Wasn't the idea to test the difference between human and machine tosses?

The link on that page doesn't resolve, nor is that what the short writeup about the undergraduate work is about.

Most of what the statisticians conclude about the physics seems to be based on poor experimental design. measuring human biases in coin flips seems to be a bit off the point-- you'd do better building a machine that emulates humans better, than to take two people and collect a large number of samples between them.

Put another way: when somebody builds a robot and collections the results of thousands of coin tosses, picking two undergrads at stanford and using their physical mechanisms does not advance any useful scientific argument about bias in coin flipping. It just muddles reality with advanced stats on heavily biased data.

Interesting experiment! Here's a suggestion for a new protocol: set up a stand at some busy places (stadiums, train stations,...) and ask random passers-by to toss a coin a few times 'for science'. Should be easier to get to statistically meaningful orders of magnitude (if not, go to a cosplay event in 'The Witcher' costumes, that should help ;)), and arguably far more representative for the situations that we care about: the two participants will have become far more experienced at coin tossing after 20k tosses, but we care more about random people who hardly ever toss coins (we already know that people who practice a lot can control the outcome of a toss with reasonable accuracy).

You are still talking about dozens/hundreds of hours of work. Better way would be to embarrassingly parallelize the problem. Get large groups of people to flip coins and collimate the result.

I'm fixated on the fact that they used a dime. That's a very small coin. I can't say I fully understand the dynamical bias mechanics but u had expected that they use a quarter. It's much larger and easier to toss.

They wanted to test the dynamics on long tosses (with lots of flipping). It is easier for a toss to be long if the coin is small.

"We adjusted the methods of the experiment for the convenience of the experimenter" is a common detail most papers leave out.

Coin tosses are not stochastic. https://www.npr.org/2004/02/24/1697475/the-not-so-random-coi...

I think you're trying to say that they are not perfectly fair, which is entirely unrelated to being stochastic.

Thank you!

No, the article actually says that sufficiently precise machine tosses can be considered deterministic. The randomness comes from humans.

> of 20,000 Heads-up tosses (tossed by Janet) 10231 landed Heads

> of 20,000 Tails-up tosses (tossed by Priscilla) 10014 landed Tails

Why not do 10,000 Heads-up and 10,000 Tails-up tosses for each person?

> > A first comment is that it would have been better for each individual to have done both "Heads up"and "Tails up" tosses (which was part of the intended protocol, but on this aspect of the protocol there was a miscommunication)

>>separate the effect of individual tossing style from any possible effect arising from the physical difference between Heads and Tails. But it is very hard to imagine any such physical effect, so we presume the observed difference (if real rather than just chance variation) is due to some aspect of different individual tossing style.

Much more boring title: "40k coin tosses reveal bad presumption and a biased coin"

I didn’t read the paper with a fine toothed comb but it seems like the two undergraduate researchers were flipping the coins by hand? If anyone can confirm or deny I would very much appreciate it! Bringing this up because it seems to introduce much variability into each flip?

Would a true rng (electron noise or whatever) be an acceptable substitute for physical coinflipping?

Has anybody tried this while under the influence of "psychic power enhancers" (psychedelics, meditation, sex, etc)?

If you do a movement 40,000 times, you will be memorizing the movement.

That reminds me of parapsychology

They have performed lots of experiments where you do a random experiment and try to change the outcome with your mind.

And this comment reminds me of one of my favorite articles I've seen linked from HN, "The Control Group Is Out of Control", relating to parapsychology as the control group for science and how that's not actually necessarily looking great for science itself: https://slatestarcodex.com/2014/04/28/the-control-group-is-o...

> It just means that the standard statistical methods of science are so weak and flawed as to permit a field of study to sustain itself in the complete absence of any subject matter.

I guess to be more fair we better flip a coin to determine which way the coin should initially face when we flip it

If we are looking that granular of detail, does the shape and weight distribution of the coin matter?

40k seems a tad on the low side, by several orders of magnitude.

Seems like a hastily done spring projects

It was an undergraduate research project.

Bring on the robot baseball umpires and the NFL robot coin flippers.

Well of course; to be anything else would be Chaos.

Referees need to start rolling a d20.

Isn’t that why you “call it in the air”?

Or just use von Neumann's debiasing algorithm - toss twice, and see if it's head-tail or tail-head, retoss when you get repeated heads or tails. It doesn't prevent dishonest tosses (if you can manipulate the bias in each toss), but should work to eliminate a consistent dynamic bias by an honest tosser.

This field of study obviously needs a much larger number of tossers.

Too bad ibankers make too much to want to be study subjects.

Next crypto class: Proof of coin tosses.

From the article

The experiment

Over the Spring 2009 semester two Berkeley undergraduates, Priscilla Ku and Janet Larwood, undertook to do the required 40,000 tosses. After preliminary experimentation with practical issues, there was formulated a specific protocol, described in detail below. Cutting to the chase, here is the complete data-set as a .xlsx spreadsheet (see sheet 2). This constitutes a potentially interesting data-set in many ways -- one could compare numerous theoretical predictions about pure randomness (lengths of runs, for instance) with this empirical data. For the specific question of dynamical bias, the relevant data can be stated very concisely

of 20,000 Heads-up tosses (tossed by Janet) 10231 landed Heads

of 20,000 Tails-up tosses (tossed by Priscilla) 10014 landed Tails

From https://www.wolframalpha.com/input/?i=binomial+distribution+... the standard deviation is almost 71, so it's a 3.3 sigma for Janet and .2 sigma for Priscilla. Since this is not particle physics, we can conclude that Janet (or her coin) is doing something wrong.

p isnt 0.5, which is the point of the article

cf. https://www.wolframalpha.com/input?i=binomial+distribution+n...

If there is the theoretical 50.8% we would expect 20,230 same-face-up in 40,000 flips. We find here 20,245. Pretty compelling.

The data is consistent with a hypothesis that there is some smaller bias in favor of the side you started with (say 50.4%) and an additional bias in favor of heads (say 50.4%).

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