Probability is just...really weird

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.

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.

I am willing to understand this, but not able to

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

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.

Beginning:😎 Middle:😕 End:😱🤯😫🤬😵‍💫🤕🧐🧐

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.

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.

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)

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

Never tell me the odds

Quite a lot of the statements is incorrect. The resolution is that the probability depends on your strategy how you will reveal. 1/2 can be correct, 2/3 be be correct, anything really. It depends on the "revelation procedure" and just saying something does not tell anything about probabilities unless you tell us how you were going to reveal in other cases.

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.

I like those odds

Isn't the paradox at the start just due to distinguishability? if you reveal one specific head, it makes the left and right head distinguishable, but if you don't, its indistinguishable?

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

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

This only makes sense if you reveal beforehand, that you will only reveal tails. Lets say the revealed hand is random, and whatever is in that hand, heads or tails, will be revealed. This results in eight cases LHT LTH LHH LTT RHT RTH RTT RHH in half of these cases the same face is revealed in the other hand.

It depends what you decided before you flipped the coins. You could say "I will tell you what I got in the left hand" or "I will tell you if I got heads in at least one hand" or huge number of other permutations. In other words this is a Bertrands paradox. Not enough information.

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.