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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

#171

Post by naniwaduni »

I_Am_Nous wrote: post #142
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 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…

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.

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

#172

Post by dfxm12 »

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 especially unintuitive if you've ever watched Let's Make a Deal, since the problem set up is oh so close to, but not exactly, the set up of the Big Deal in the show. It's too easy to conflate the rules of the show with the math problem, which will lead to confusion.

I think seeing the results of a simulation also elucidates the set up of the math problem vs reading a proof.

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

#173

Post by nofinator »

098799 wrote: post #84
ChocMontePy wrote: post #69

Earlier quoted context omitted.

If anyone wants to test it, someone wrote a short code that simulates doing that 100,000 times: https://www.techiedelight.com/generate-fair-results-biased-c... The coin is biased to come up TAILS 80% of the time, but using Von Neumann's method in the program I got HEADS 50.035%, TAILS 49.965%.

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

Ironically, this reminds me of a story (folk tale?) about Von Neumann himself.

A colleague told him about the Two Trains Problem (https://mathworld.wolfram.com/TwoTrainsPuzzle.html), and Von Neumann replied with the correct answer. When his colleague said, "Ah! You figured out he trick!", Von Neumann replied, "What trick? I just summed up the distances in my head!"

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

#174

Post by BurningFrog »

098799 wrote: post #84
ChocMontePy wrote: post #69

Earlier quoted context omitted.

If anyone wants to test it, someone wrote a short code that simulates doing that 100,000 times: https://www.techiedelight.com/generate-fair-results-biased-c... The coin is biased to come up TAILS 80% of the time, but using Von Neumann's method in the program I got HEADS 50.035%, TAILS 49.965%.

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

You're of course right, but maybe 1% of the population understands that, while 100% understands the practical test.

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

#176

Post by slingnow »

bunderbunder wrote: post #129
cabirum wrote: post #61

50 authors in a paper about flipping coins?

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.

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

#177

Post by jszymborski »

098799 wrote: post #84
ChocMontePy wrote: post #69

Earlier quoted context omitted.

If anyone wants to test it, someone wrote a short code that simulates doing that 100,000 times: https://www.techiedelight.com/generate-fair-results-biased-c... The coin is biased to come up TAILS 80% of the time, but using Von Neumann's method in the program I got HEADS 50.035%, TAILS 49.965%.

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 folks have more faith in their ability to derive proofs than write simulations and vice-versa.

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

#178

Post by aidenn0 »

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.

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

#179

Post by 4star3star »

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 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 a 2/3 chance of picking a Goat, so, effectively, you want to pick a Goat so that you switch to the Car.

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

#180

Post by jackfoxy »

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 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.

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