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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

#231

Post by s1artibartfast »

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…

I always feel like there is something fundamental missing from the examination of Monty Hall problems.

I think it has to do with the difference between "probable outcome in reality" and "probably outcome based on personally known information".

Lets say when you get down to doors #1 and #49, Monty brings in someone new, with no information and says pick a door. For that new person, standing right next to you, doors #1 and #49 have a 50-50% chance, while for you they are a 2% vs 98% chance.

How can door #1 simultaneously have a 2% chance for you and a 50% chance for Bob? The answer is that the chance is not a single fixed property of the door itself- which is hard to wrap ones head around.

And for that matter, Monty Hall himself knows one of the doors is 100% and the other is 0%.

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

#232

Post by mucle6 »

hmate9 wrote: post #29
brettwall wrote: post #23

Based my back-testing of stock market, I found a similar conclusion: if a stock rise yesterday, then today the probability of raise > the probability of fall. P(raise) is about 50.1%, P(fall) is about 49.9%. Vice versa. Having this theory means you can't rely on a single bet, you have to bets many many times to make profit from stock market. Even though I knew that, I am still working as a developer, I wish one day I…

You have to take into account how much it rises and how much it falls too. It might be you win more often but when you lose you lose more.

I don't know enough, but I bet with options you could make something close to a double or nothing boolean bet

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

#234

Post by Natsu »

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…

This isn't so mysterious once you learn a bit of do calculus and realize that observation and intervention are fundamentally different things.

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

#235

Post by Natsu »

s1artibartfast wrote: post #217
naniwaduni wrote: post #171

Earlier quoted context omitted.

Fundamentally, the trouble with the Monty Hall problem isn't that analysis comes to the wrong answer, it's that people often come to the wrong model when reasoning about it informally. It's not any harder to do the "correct" analysis than to write up a simulation. It's mostly just easier to convince yourself that the simulation matches the problem description when it reaches the unintuitive result.

Fundamentally, I think the real trouble with the Monty Hall problem is that the assumptions of the game are not clearly stated. Because of this, people come up with different models.

This modelling ambiguity is resolved by do calculus, which makes a clear distinction between intervention and observation: https://arxiv.org/pdf/1305.5506.pdf

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

#237

Post by stainablesteel »

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_...

you can do that all you want my coin is the same on both sides

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

#238

Post by Natsu »

s1artibartfast wrote: post #231
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 always feel like there is something fundamental missing from the examination of Monty Hall problems. I think it has to do with the difference between "probable outcome in reality" and "probably outcome based on personally known information". Lets say when you get down to doors #1 and #49, Monty brings in someone new, with no information and says pick a door. For that new person, standing right next to you, doors #1…

There is something missing: regular stats don't differentiate between doing things and observing things and these two are not at all the same. If I have a digital thermometer and I observe it to show a high temperature, then I will note an association between that and feeling warm. But if I merely set the thermometer gauge to a high value artificially, it's not going to make me feel any warmer.

This ambiguity is resolved by something called do calculus - https://arxiv.org/pdf/1305.5506.pdf

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

#239

Post by Natsu »

voidfunc wrote: post #88
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

Some of us suck at math.

Part of the problem is that the basic statistical model simply neglects to differentiate between observing and doing, which changes the odds. This is very important when trying to reason about causality. When you observe an association like your thermometer shows a high number when it's warm out it's one thing, but when you set your thermometer to a high number you won't get any warmer. Whereas if you warm the room, your thermometer will rise. This symmetry breaking is captured by something called do calculus.

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

#240

Post by Q_is_4_Quantum »

cantrevealname wrote: post #182

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…

Perhaps it helps to imagine someone had a "screwy thumb" and the coin only precesses when they "flip" it (in fact people can train themselves to do this, and its very difficult for you, the sucker, to see in the air that the coin is not rotating but just precessing!). Hopefully its obvious that whatever side is initially facing up will be the same one facing up when its caught?

The next step is not at all intuitive to me, namely that even someone trying to do a fair flip causes some precession, and that this isn't decoupled from the rotation.

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