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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

#251

Post by cortesoft »

bigstrat2003 wrote: post #249
benchaney wrote: post #247

Earlier quoted context omitted.

The best probability estimate you can make is constrained by the information you have available. The new person showing up has less information than the existing constant, so it makes sense that their best estimate would be less precise. Similarly, if someone with x-ray vision walked up in the middle of the game, they could pick the car 100% of the time, because they have access to more information than either of the…

I don't see how a new contestant has less information, though? They know that one of the two doors contains the prize, which is all the previous contestant knows either.

The crucial bit of information that the new contestant doesn't have is that there was a door that was ineligible to be eliminated (the door chosen by the original contestant).

If the game had different rules, it would work like you are imagining. Specifically, if Monty randomly eliminated one of the two doors, meaning there was a chance for Monty to reveal the prize instead of a goat. If Monty has the chance to eliminate the prize before giving the contestant a chance to switch, then switching does not give you an advantage.

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

#252

Post by vince3455 »

alexdowad wrote: post #245
s1artibartfast wrote: post #217

Earlier quoted context omitted.

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.

That's absolutely right; further, if you explicitly model the behavior of the game show host, you can exhibit models under which "it's better to switch" and models under which "it doesn't matter if you switch or not".

>models under which "it doesn't matter if you switch or not".

Could you provide an example? It seems obvious that a switcher wins exactly when a non switcher looses, which is 2 / 3 ?

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

#253

Post by Q_is_4_Quantum »

Given access to repeated uses of a coin of unknown bias "p" (which is not 0 or 1) you can (eventually) always generate a new coin flip with bias given (exactly) by:

1. 1/2 (i.e fair - von Neumann)

2. p^2

3. p^2/(p^2+(1-p)^2)

4. sqrt(p)

Number 4 really surprised me, I learned it from this paper: http://www.math.chalmers.se/~wastlund/coinFlip.pdf

But you can never generate the biases:

5. 2p

6. 4p(1-p)

Although... if you change the game to allow a quantum coin then 5. and 6. are possible (a paper of mine: https://arxiv.org/abs/1509.06183)

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

#254

Post by dfxm12 »

bigstrat2003 wrote: post #243
cortesoft wrote: post #195

Earlier quoted context omitted.

Yeah, I feel like the Monty Hall confusion goes away if you are explicit about the rules: "Hall will always open one of the two non-chosen doors and will never reveal the prize" I think most people who don't understand the problem miss that critical detail.

No, I don't think that detail makes it any easier. I know that but I still really can't accept the correctness of the Monty Hall strategy (I have to basically just take it on faith and stop trying to understand it). I was trying to put my finger on why, and I think it's this. After Monty eliminates one of the three doors, then the prize is behind one of the two. If someone were to come in this point, with no prior kn…

Maybe this will help understand it intuitively. You have a choice between doors 1 2 3. You pick door 1. You know the odds of the car being in door 1 is 1/3. The odds of the car being in door 2 or door 3 are 2/3.

Monty opens door 3, showing a zonk. You knew there was a 2/3 chance of the car being in door 2 or 3, but now you know there's a 2/3 chance of the car being in door 2 (since you know it is not in door 3).

All this didn't change anything you know about door 1. It has the same 1/3 chance it started with. Probability is all about what you know in the moment.

The math involves understanding the rules, that Monty will never open the door you picked and will never open the door with the car behind it. This is why one can't look above and say "well, there is a 1/2 chance of the car being behind door 1 after door 3 was opened and there wasn't a car there". This would only be true mathematically if the door Monty opened was random, but we know the door Monty picks isn't random. In fact, the pool of doors that could be opened depends on your initial pick. Monty was never going to open door 1 (the door that you picked), even if it was a zonk & Monty was never going to open the door with the car, therefore one can't make that assertion.

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

#255

Post by benchaney »

bigstrat2003 wrote: post #249
benchaney wrote: post #247

Earlier quoted context omitted.

The best probability estimate you can make is constrained by the information you have available. The new person showing up has less information than the existing constant, so it makes sense that their best estimate would be less precise. Similarly, if someone with x-ray vision walked up in the middle of the game, they could pick the car 100% of the time, because they have access to more information than either of the…

I don't see how a new contestant has less information, though? They know that one of the two doors contains the prize, which is all the previous contestant knows either.

When one door was opened it revealed information about the other two doors.

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

#257

Post by alexdowad »

vince3455 wrote: post #252
alexdowad wrote: post #245

Earlier quoted context omitted.

That's absolutely right; further, if you explicitly model the behavior of the game show host, you can exhibit models under which "it's better to switch" and models under which "it doesn't matter if you switch or not".

>models under which "it doesn't matter if you switch or not". Could you provide an example? It seems obvious that a switcher wins exactly when a non switcher looses, which is 2 / 3 ?

Take a game show host who lets you choose a door, randomly reveals what is behind one other door, and then gives you an opportunity to change your choice. This game show host CAN (randomly) reveal the prize; he has equal probability of revealing ANY of the unchosen doors.

Say you are playing the Monty Hall game with this host. You choose your door, he opens another door, and it happens (purely by chance) that there is no prize there. Do you still believe that you have a 2/3 chance of winning if you switch to the other unopened door?

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

#258

Post by gowld »

nomilk wrote: post #83
fbartos wrote: post #2

About a year ago, we embarked on a quest to answer one of the most intriguing questions: If you flip a fair coin and catch it in hand, what's the probability it lands on the same side it started? Today, we are finally ready to share the results. Thanks to my friends, collaborators, and even strangers from the internet, we collected flippin 350,757 coin flips. We ran several "Coin Tossing Marathons" (e.g., https://you…

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…

Bouncing erases the bias due to flipping.

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

#259

Post by bryanbuckley »

gala8y wrote: post #6
fbartos wrote: post #2

About a year ago, we embarked on a quest to answer one of the most intriguing questions: If you flip a fair coin and catch it in hand, what's the probability it lands on the same side it started? Today, we are finally ready to share the results. Thanks to my friends, collaborators, and even strangers from the internet, we collected flippin 350,757 coin flips. We ran several "Coin Tossing Marathons" (e.g., https://you…

This was my intuition in childhood. If you choose tails to be yours and start with tails then catch it, it is most likely to be tails. I came up with this observation myself. Weird.

Noticed as a kid I could flip a quarter with a certain consistency, so I experimented a bit and quickly got to be >90% accurate with an ordinary (controlled) flip.

Pretty simple. In fact I just picked up a quarter and practiced (20+ years out of practice) and have some observations: 1) harder than when I was a kid, my fingers are lot bigger + stronger so it's not as precise from the start. A bigger and heavier coin would help. 2) the timing factor is bigger than I recalled.. essentially you can watch the coin flipping and get a subconscious/automatic/predictable sort of count/feedback to it. You can bring your hand up to the coin in the air at a precise moment pretty easily and "tell" (>90% accuracy today of the flips I just did that I considered successful before looking at the result) if the flip was predictable. Hand eye coordination, spatial awareness is very correlated to this skill, I suppose. 3) it really is the same side that comes up.. again I think because of the automatic watching/count/completion of full rotations, i.e. catching the coin at the end of a full rotation instead of a partial.

Came in handy occasionally.. if I knew I was going to be wrong (other person usually waits to call mid-flip) I could catch the coin a little lower to give myself a chance, or punk them by not putting it on the back of my hand as is more standard (they might demand a re-flip.. kind of like if you are playing rock paper scissors and one person goes on 3 and the other on 4).

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

#260

Post by gowld »

matteoraso wrote: post #256

I actually used to be really good at manipulating this as a kid. Basically, if you toss a large coin with a stiff arm, you can get it to flip exactly 1 and a half times before you catch it. I would always use this to win bets with my friends.

If you are allowed to cheat, you can also spin the coin to not flip at all.
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