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Fair coins tend to land on the same side they started

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Re: Fair coins tend to land on the same side they started

#201

Post by cstrahan »

matsemann wrote: post #198
tacitusarc wrote: post #186

Earlier quoted context omitted.

For sufficiently analytical folks that works, but for lay people it tends to still be confusing. The best way I’ve heard it explained to help people get it through intuition is by changing the number of doors and goats. Say there are 100 doors, and they all have goats except one, which has a car. You pick door 1. Monty then proceeds to open doors 2 through 48, skips door 49, and then opens the remaining doors. After…

I've never been happy with that explanation. I don't get why the host would not just open a single door, that's what the host does in the other scenario to me.

> that's what the host does in the other scenario to me.

Is it, though? It seems apparent that, after the first guess, the host opens all but the last two doors, which just so happens to be 1 door.

To check the math:

Start with the $NUM_DOORS open doors. Now open all but the last two. So that’s $NUM_DOORS-2, which is 3-2, which equals 1 open door.

Re: Fair coins tend to land on the same side they started

#202

Post by aidenn0 »

elijaht wrote: post #181
aidenn0 wrote: post #178

Earlier quoted context omitted.

That is only guaranteed to work if subsequent tosses are independent of each other. TFA suggests that they are not. [edit] If you always start with the same side of the coin face-up, then the tosses will be independent of each other, but if you e.g. always flip it once or always keep it the same before the next toss, then they are not.

TFA does not suggest that. Even if a coin is biased towards the side it started on, this wouldn’t carry over from one flip to the next. It would be important to start the flip on the same side, but that’s doesn’t make the second flip dependent on the first

Hah, realized that after I posted and edited while you were commenting. It's a good point. My original assumption would you'd just pick it up and flip it; not attempt to put the same side face up each time.

Re: Fair coins tend to land on the same side they started

#203

Post by qup »

kortex wrote: post #141
fowlie wrote: post #130

Very interesting! Is there reason to believe that the outcome would be different if the experiment was re-run but replacing the humans with coin-flipping machines?

Folks have made flipping machines than can achieve near perfect reliability. Same guy mentioned in the paper. https://www.npr.org/2004/02/24/1697475/the-not-so-random-coi...

Sorry, does this mean perfect 50/50, or 100% controllable?

Re: Fair coins tend to land on the same side they started

#204

Post by strangattractor »

cantrevealname wrote: post #128
098799 wrote: post #84

Earlier quoted context omitted.

Why would you test it? Probability of two heads: p*p Probability of two tails: (1-p)*(1-p) Probability of head followed by tails: p*(1-p) Probability of tails followed by heads: (1-p)*p It's not difficult to notice that if you remove the first two, the last two form a 50/50 distribution

> Why would you test it? I recall conversations on Usenet decades ago about the Monty Hall problem[1] in which people gave elementary proofs that probabilities don't change by opening a door. Even from mathematicians and statisticians. People were very insistent that the analytical solution was simple and obvious and that switching doors didn't change anything. The only thing that changed some people's minds was a pr…

The Monty Hall problem is small enough to enumerate all the out comes on paper. If you then test all the different scenarios (not many) and compare switching to not switching - switching produces a slight edge. So it can be done without a computer and not a great deal of effort. I am not a mathematician so it was the only way I could prove it to myself at the time.

Re: Fair coins tend to land on the same side they started

#205

Post by mensetmanusman »

jackfoxy wrote: post #180
cantrevealname wrote: post #128

Earlier quoted context omitted.

> Why would you test it? I recall conversations on Usenet decades ago about the Monty Hall problem[1] in which people gave elementary proofs that probabilities don't change by opening a door. Even from mathematicians and statisticians. People were very insistent that the analytical solution was simple and obvious and that switching doors didn't change anything. The only thing that changed some people's minds was a pr…

The Monty Hall problem is a fun one to code up, and yeah, there are otherwise smart people who refuse to believe it. I coded it up in F# https://github.com/jackfoxy/LetsMakeADeal to convince one of the founders of a start-up I worked for. He just grunted and walked away. Pretty sure he still doesn't want to hear about Bayes' Theorem.

Just start with an infinite number of doors and move backwards from that.

Re: Fair coins tend to land on the same side they started

#206

Post by dotancohen »

pmontra wrote: post #149
I_Am_Nous wrote: post #142

Earlier quoted context omitted.

The Monty Hall Problem is a fun one because you can try to approach it from a purely analytical perspective and get one answer, while incorporating the whole situation (especially the fact that the final probability is not natural as they force the final decision into far fewer doors than originally present) and testing you can find a different answer entirely. I suppose this is an interesting corollary with discover…

That's how it went when I was solving problems at the Statistics course at university. I modeled the problem perfectly, got the wrong result. Changed assumptions, got the wrong result. Checked the solution, its reasoning didn't make much sense anyway. Run a simulation, got an approximate result close to the correct solution.

This sounds like the classic "tweak the model until the results fit with our preexisting conclusion". Very common across all industries unfortunately.

Re: Fair coins tend to land on the same side they started

#207

Post by aspenmayer »

pirates wrote: post #120
aspenmayer wrote: post #71

Earlier quoted context omitted.

Your comment reminded me of two-up. > Two-up is a traditional Australian gambling game, involving a designated "spinner" throwing two coins, usually Australian pennies, into the air. Players bet on whether the coins will both fall with heads (obverse) up, both with tails (reverse) up, or with a head and one a tail (known as "Ewan"). The game is traditionally played in pubs and clubs throughout Australia on Anzac Day,…

Two-up sounds pretty fun. Your comment in turn made me think of Chō-han, which is somewhat similar but involves rolling dice instead of flipping coins. https://en.m.wikipedia.org/wiki/Ch%C5%8D-han

It’s wild and raucous and usually takes place outside pubs and local workers’ clubs, lawn bowl clubs etc. The spinner puts the coins on a flat stick made for the game, but basically a popsicle stick but wider like a tongue depressor. They toss the coins up and flick the stick to tumble the coins. If you called the two matching coins correctly, you double your bet in winnings. Each game takes like 30s-1m and they go on from mid-morning til early afternoon ish.

The losers indirectly pay the winners based on your call of two heads or two tails and if there’s a split, the house wins that game.

Most of the time is spent drinking your beverage of choice, yelling and cursing your own luck and talking smack to the coins, the spinner being booed or cheered for a well run game or a bad string of luck, but the bets are typically fairly low in my experience, although it’s up to each individual player what they bet, but the spinner or venue sets the bet amount per section or per spinner. Each spinner usually takes a set amount, like $5/$10/$20/$50, even $100 in some cases. It’s all cash and pretty much the honor system in that they aren’t handing out receipts or tickets with your bet, so that sets an upper limit on how many bets each spinner can keep straight. No one wants to see someone lose their shirt, so most folks play against their friends/mates for fun, and ultimately you’re all playing your own game because you decide your bet and it’s actually fairly unpredictable due to the crowds milling around the spinners, of which there will be many, all taking bets and running games independently and simultaneously, and players can place bets in any or all of the games around them if they want.

It’s a wild affair. Highly recommended.

Re: Fair coins tend to land on the same side they started

#208

Post by sixstringtheory »

fbartos wrote: post #102
nomilk wrote: post #83

Earlier quoted context omitted.

It becomes clear why there's a same-side bias when watching the video: https://www.youtube.com/watch?v=3xNg51mv-fk These are fairly gentle coin tosses; barely going a foot into the air! When I think of a coin toss, I think high and spinning fast (like the ones before sports games, where the coin goes into the air and lands on the ground, usually rolls a short way, and is collected on whatever side it landed). I would…

There was indeed a lot of variation in the height of the tosses. I however disagree with the conclussion: two of my friends at the video had the most different height of tosses (one tossed thrice as hight as the other one), yet both of them had exactly the same bias (0.505). The amount of spin is unfortunatelly very misleading from the 30fps videos--the coins often seem like not spinning at all but that's just a resu…

How do you control for bias coming from the same coin flipper? Do they usually flip their coin from the same starting height (whatever comfortable arm positioning they have, which I assume would also introduce bias by how they catch it as well) and to the same arc peak height? Or were they encouraged to try a different body position, strength and angle of launch for each flip?

Re: Fair coins tend to land on the same side they started

#209

Post by aplusbi »

4star3star wrote: post #179
cantrevealname wrote: post #128

Earlier quoted context omitted.

> Why would you test it? I recall conversations on Usenet decades ago about the Monty Hall problem[1] in which people gave elementary proofs that probabilities don't change by opening a door. Even from mathematicians and statisticians. People were very insistent that the analytical solution was simple and obvious and that switching doors didn't change anything. The only thing that changed some people's minds was a pr…

The way the problem makes sense to me is this. Doors: Goat Goat Car You pick a door. Monty shows you a Goat. You switch or stay. Monty will never show you the Car before offering a switch. He always shows you a Goat. It doesn't matter which Goat he shows you - it's just "not the Car". If your first choice is a Goat, switching will win you the Car. If your first choice is a Car, switching will win you a Goat. You have…

The best way that I've thought about it is like this:

You pick a door, then Monty let's you switch to the two remaining doors and if the car is behind either of them you win.

Obviously choosing the two remaining doors is better.

The trick is to realize that Monty showing you the contents of one door and letting you choose the other one is identical to Monty letting you choose both the remaining doors.

Re: Fair coins tend to land on the same side they started

#210

Post by rootusrootus »

misja111 wrote: post #93
helsinkiandrew wrote: post #85

I think you can argue that the experiment wasn't representative of 'normal' coin flips. On average, each flipper in that experiment flipped a coin over 7000 times, after that amount many people will have learned to flip in a comfortable way with less variance between physical action and force they use. I'd imagine that in that case the coin would more likely land with the same orientation. I don't think this would be…

I bet that with enough practice, you can learn to toss the coin so that you get a much bigger than 51% chance of landing it on the same side.

I think you'd need more than practice. Most people would need a teacher. Just doing something over and over isn't automatically going to make you better. E.g. The old "10000 hours of practice makes you an expert" rule assumes deliberate practice. And even then, it's incorrect.
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