Probability is just...really weird

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[leftylizard]

"Wait, it's all the Monty Hall Problem?"

"Always has been"

[Riko]

He said that both of them were heads, and don't believe the internet. I think that statement contained a lie and he actually didn't have any coins in his hands.

[LumberJackJR]

I am willing to understand this, but not able to

[hotelmasternm]

I'm watching it with no sound and I'm just imagining "the odds I beat you with this fist...but the odds I beat you with this fist...either way you're getting these fists"

[HeavyMetalMouse]

It seems like, ultimately, the difference comes down to the conditions placed on the information.

One of the key aspects of the Monty Hall phenomenon is the fact that the host is not giving you a wholly arbitrary piece of information, but specifically giving you a piece of information about a door that has a donkey. The confidence in which door has the prize changes because we know that he is *limited* to only opening a donkey door. If we did not know that information - that is, if we believed that he was simply opening one of the two remaining doors at random to show us whatever happened to be behind it, before asking us if we want to switch, then the actual information we get from the door opening is completely different: if he opens the prize door, we know what door the prize is in; if he opens a donkey door, we know that one of the two remaining doors must have the prize with equal probability. Because Monty does not know nor care what is behind the door he is opening in this altered formulation of the problem, it does not create the same result as the normal problem.
Essentially, if you did not know that Monty would *never* open a door with the prize showing, the reveal of the donkey would change your confidence differently than if you *did* know that fact.

As such, if, before flipping the two coins, you say "If I flip at least one tails, I will reveal to you one of the coins that flipped tails, and you will have to guess the other one." and thus, the statement would *only* be made if there is at least one tails being flipped, that statement now contains different total information than if they had not prefaced the reveal with that statement. The entire process of updating our confidence depends on whether or not we know what preconditions are placed on giving us that information.

Calling these confidences 'probabilities' only serves to make it seem like something weird is 'happening', when in actuality, nothing is happening. The status of the things in play have not changed, only our level of knowledge about them - as such, it makes more sense to speak in terms of how 'confident' we are about whether or not a given state is the one we are observing, given limited and conditional information, and recognizing that the conditions under which the information is given is part of the information. Formulated in that way, it becomes fairly reasonable to expect that, given different information (including the conditions of that information), we would have a different level of confidence in a given result being true. The fact that our confidence in a given result follows analogous rules to the mathematics of probability is more down to the definition of confidence in the context of Bayesian Inference, which attempts to make a mathematically rigorous idea out of the concept of confidence in an unknown and thus uncertain outcome.

[undergroundhead]

Much more intuitive version of the "2 daughters/one named Julie" problem you covered a few years ago

[OMGclueless]

This is at the heart of a deep problem in science right now called "p-hacking". In sciences that use random sampling, it's generally accepted that you need a result with p < 0.05 in order to publish a paper. This statistical measure "p" is supposed to mean there is only a 5% chance of seeing this result in a random sample if the hypothesis is false. But just like this demo where how the presenter chooses the hand to reveal determines how to interpret the result, in science if the experimenter has lots of ways to slice up their data set and lots of possible analyses they can do looking for "statistically significant" results then revealing the positive result (just like revealing heads in some hand) is less significant than the p-value suggests.

[BenAlternate-zfnr]

It's not obvious that the experimental procedure is "if one coin is tails, reveal one tails". It could also have been "choose one coin at random and reveal it" (this setup does indeed yield 50% in both the light-out and regular versions).

In the version where you just reveal the existence of a tail without identifying which coin it is, the answer again depends on your policy for revealing information. For example, if you always reveal the total number of tails, then there is 0% to be two tails when you say there is at least one.

[weremuppet]

There's another way to look at it when it comes to probability as well, it's to look at the 2 coins as 2 separate datasets
Coin A
Heads
Tails

Coin B
Heads
Tails

If you reveal Coin A as Tails, you have removed that dataset from the equation (more or less a 2 option monty hall) and there's a 50/50 chance of coin B being Tails as well
If you reveal Coin B as Tails, you have removed that dataset from the equation (more or less a 2 option monty hall) and there's a 50/50 chance of coin A being Tails as well

If you say that one of the datasets are Tails, you are left with 2 separate but dependant datasets that each have a 50/50 chance of being Tails (if coin A is tails then Coin B 50/50, if Coin B is Tails then Coin A 50/50)

[noahsammond]

I hope that Zach is able and willing to do more math videos

[ethandavis]

It's important to rigorously define the procedure. It's not enough to just assume that one would have equal probability of choosing either hand in a tails-tails scenario.

[jonathanlevy]

3:29 a quick correction: not necessarily those last two TT occurwith the same probability because they depend on your choice (which the viewer don't know) but in any case the sum up to 1/3

[LuigiGamingLounge]

I wrote a comment on your last video (the one about the three obvious questions that actually aren't) I wrote a lot about this last question, and how saying that the probability changes when one is revealed from when you just knew that one of them was tails had some faulty reasoning. Then I was recommended this video (which was made 2 weeks ago at the time) and I watched through this video and was glad to see your clarification of how the "probability" in a mathematical sense changes, but in in a "probability" sense. As strange as that sounds, thank you for making this follow-up video. Very good work, brother!

[jaap_vink]

The probability doesn't change. The information changes and therefor the probability space.

[colbyforfun]

It matters that the person asking the question knows both coin identities, and chooses to only reveal one. Technically we don't have enough information because we don't how they decide which coin to reveal, or whether there is a chance they will reveal a heads instead. Because Zach already has all the information and is selectively revealing some of it, no new information is actually discovered. If Zach instead flips two coins and checks only the left one, revealing it to be tails only as he learns it is tails himself, then the odds do jump to 1/2 because new information is discovered.

[kurtclark]

A different way of thinking about it, is when you say “this one is tails’, you are essentially saying ‘this one is tails and I have one other coin’ or ‘I have one (unknown) coin, and the odds being being 50/50 are obvious.
If you only say “one coin is tails”, you cannot make the reduction.

[OBCOR]

That's an important point to consider when dealing with probability. The same thing also applies to the Monty Hall problem - the "2 out of 3 should switch" solution only works if you know the host's behavior in all cases.

[gregaldr]

I didn't think you understand it yourself. There are two ways to play this game. In one case, that two coins are considered equivalent (interchangeable) which means for example that TH=HT and the are exactly 3 possible outcomes. However, another way to play this is to add the notion of position such that TH != HT. I'm this case, there are exactly 4 possible outcomes. For the game to make any sense, this must be clearly articulated at the beginning. There is no paradox as you are simply not specifying the equivalence (or non-equivalence) of the coins) up front and then conflating the two cases to present the APPEARANCE of a paradox where none exists. Perhaps this was intentional but if so the results are meaningless and the game uninteresting.

[ianfowler]

This is very similar to Monty Hall. After the contest picks thier door, then Monty is confronted with ONLY 2 possibilities. Goat, goat or car, goat. Thing is these 2 outcomes are NOT equally likely as Monty will see goat, car twice as often as goat, goat. Nice video.

[franksierow]

In the beginning part of the video, there is a dependence on counting possibilities. You cannot go straight from this *counting* to calculating *probabilities* without showing that all the possibilities are equally likely.