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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

#181

Post by elijaht »

aidenn0 wrote: post #178
jbandela1 wrote: post #16

Von Neumann described a very elegant way to get fair results from a biased coin. 1. Flip the coin twice 2. If you get the same result both times, goto 1 3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result. https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...

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

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

#182

Post by cantrevealname »

I'm still looking for an intuitive or ELI5 explanation of the mechanism for this bias.

The original paper says:

The standard model of coin flipping was extended by Persi Diaconis who proposed that when people flip a ordinary coin, they introduce a small degree of precession’ or wobble—a change in the direction of the axis of rotation throughout the coin’s trajectory. According to the Diaconis model, precession causes the coin to spend more time in the air with the initial side facing up. Consequently, the coin has a higher chance of landing on the same side as it started.

Another coin toss experiment[1] site says this:

The basic reason is that, instead of rotating around a horizontal axis as one might imagine, a typical tossed coin is rotating around a tilted axis which is precessing in 3-space, and this entails a certain degree of "memory" of the initial parameters.

The Diaconis paper[2] has the definitive explanation but it's hardly intuitive. I got a feel for why it is, but I can't do an ELI5. The best I'm able to write is this: A human being is likely to introduce some precession in the coin toss. If there is precession, then the angular momentum vector is going to spend more time in the heads direction if starting from heads, and that accounts for the bias.

What I think would work well for an ELI5 is an animation of a coin toss showing the angular momentum vector sweeping out a region during its flight, and showing it spends slightly more time pointing toward heads.

[1] https://www.stat.berkeley.edu/~aldous/Real-World/coin_tosses...

[2] http://epubs.siam.org/doi/10.1137/S0036144504446436

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

#183

Post by k7sune »

jbandela1 wrote: post #16

Von Neumann described a very elegant way to get fair results from a biased coin. 1. Flip the coin twice 2. If you get the same result both times, goto 1 3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result. https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...

How come when the results are the same you have to go to 1 and flip twice again? Can you just toss another one and use the last two results?

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

#184

Post by bunderbunder »

slingnow wrote: post #176
bunderbunder wrote: post #129

Earlier quoted context omitted.

Make me sit around flipping coins 10,000 times and recording the results, and you damn well better at least put my name on the paper.

Then I guess we should include every study participant as an author across all disciplines. 200 participant study in psychology? 200 authors on the paper.

Not necessarily? The difference is that the people flipping the coins are not just experimental subjects, they're actually implementing the experimental protocol.

The kinds of participants you're talking about are not just left off the paper out of lack of interest. It's often the ethically preferable option. They often have a vested interest in remaining unnamed for privacy reasons, and derive no tangible benefit from being listed as authors.

A big author list isn't totally unheard of. The paper where they announced the discovery of the Higgs boson had an author list that spanned 8 densely-packaged pages.

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

#185

Post by nullc »

jbandela1 wrote: post #16

Von Neumann described a very elegant way to get fair results from a biased coin. 1. Flip the coin twice 2. If you get the same result both times, goto 1 3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result. https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...

The VN debiaser is very simple but it's not very efficient-- it loses a lot of your randomness.

Under the same IID assumption you can take N flips that returned M heads and map them to the N choose M possible ways that could have happened. The result will (under IID assumption, even in the presence of bias) be a uniform number on the range [0..N choose M). The ctz(N choose M) trailing bits can be used directly (as they will be uniform) but the rest would have to be converted to binary via something like an arithmetic coder or rejection sampling.

The result is muuch more efficient.

Less directly, VN debiasers can also be stacked. Each debiaser outputs three streams: the normal one, one that says if the normal one output anything, and one that says if it got HH or TT. Then run VN debiasers on those. Though it takes a fairly large tree to extract most of the entropy.

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

#186

Post by tacitusarc »

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…

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 all that, he stops and asks you, would you like to switch?

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

#187

Post by bscphil »

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…

I think the great advantage of "simulation", for the programming-literate, is not that you can simulate your way to a correct answer, but that the process of creating a simulation is likely to show you the error in your reasoning.

As a young teenager, I encountered the Monty Hall problem for the first time, and I didn't believe that the "analytical" answer was correct. I decided to simulate it by programming. In the 20 minutes it took me to write a simulation, I went from complete incomprehension to a full understanding of why I got the results I got. Programming a simulation of the problem forces you to write out the algorithmic significance of "Monty reveals one of the goats".

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

#188

Post by thaumasiotes »

tacitusarc wrote: post #186
4star3star wrote: post #179

Earlier quoted context omitted.

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…

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…

There's a better way to think about it.

1. You pick a door.

2. You get the offer "Do you want to keep that door, or choose both [all] of the other doors? In either case, you'll keep anything that isn't a goat."

3. Nobody opens any doors.

Should you keep your one door, or switch to the two doors?

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

#189

Post by thaumasiotes »

jbandela1 wrote: post #16

Von Neumann described a very elegant way to get fair results from a biased coin. 1. Flip the coin twice 2. If you get the same result both times, goto 1 3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result. https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...

That only works if the result of the coin is independent of whatever it showed on the prior flip.

(You might say "of course it is!", but if that's your approach to the problem, you should be aware that biased coins don't exist...)

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

#190

Post by shapefrog »

tiagod wrote: post #77
shapefrog wrote: post #25

Can someone who has access to a precision robot and controlled environment please check; if one applies the same force on the coin in flipping, with it landing on a (soft) surface at the same height it - will it land the same side every time.

Your comment sniped me into thinking of some over-engineered systems to measure this with the human factor included. Some sort of RFID coin balanced on all axis with an IMU chip that measures the force applied to the force (through acceleration) and the mode and number of rotations of the coin. Maybe a computer vision solution would also work

Moving hoop won't let you miss - https://www.youtube.com/watch?v=myO8fxhDRW0

I guess in theory you could "catch" with the correct call by varying the height of the catching platfrom.

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