Did you do this yourself ?
Or are you also "reasoning with math" ... or without it ?

I said you're a hopeless case, but as sometimes you do make good points, I cannot wrap my mind around you being confused to that point.

People running races are doing *cumulative* work.  If you run a race, you cannot win the race at your first step.  You have to do a certain number of steps.  This is why this is a bad analogy with mining, where each hash is an independent lottery draw.  People running do not "draw a lottery of winning" at each step they take in the race.  The distribution of winning of race runners is strongly peaked around "10 minutes".  It is not an exponential distribution.
A guy just starting out and doing his first step has much less chances to win the race, as compared to a guy that has already run 90% of the distance.  While a miner calculating one single hash has just as much chance of winning a block than someone who has been hashing for the last 50 minutes, at the next hash.

If you want a better analogy, take sets of 5 dice, and consider that the difficulty is "throwing at least 4 times a six out of 5 dice".

Suppose that you are throwing your 5 dice every 3 seconds.
Suppose that others are doing the same with their dice, but some throw them every second, others, only every 5 seconds.  (they have different hash rates)

Whenever one of you "hits" a 4 six out of 5, he "wins a block".  It is only when two of you happen to win within the same second, that the first one wins, and the second one loses his won block.

Think of how "the others winning" influences the rate at which you win, and how many times you lose because of competition.

You should understand what "mining" is: it is finding a hash of a given block that satisfies a "rareness" condition: the difficulty.  If the difficulty is, say, 1000, that means that only one hash out of a thousand satisfy this requirement.  Now, the specificity of a cryptographic hash is that you cannot have the slightest idea of what it is, before you've calculated it.  This means that each time you calculate a hash of something, you have one chance out of 1000 to have a hash that satisfies the condition.  But it is really random.  It could be the very first hash you calculate, and it could be only after you've calculated 2000 of them.    However, ON AVERAGE, you will win one good hash every 1000 hashes you calculate, because, exactly, one hash out of 1000 is a good one.

Now, your hardware can calculate a certain number of hashes per second: it is your "hashrate".  If you can calculate, say, 20 hashes per second, then you will need 50 seconds to do 1000 hashes.  So ON AVERAGE, you will win a good hash every 50 seconds.  But it can be after 1 second, or it can be after 200 seconds, because the lottery is random.  However, ON AVERAGE you win a block every 50 seconds.

Now, bitcoin (and many other PoW crypto) have a self-regulating difficulty, so that the difficulty INCREASES until the average time of winning a block is 10 minutes for the whole network.  However, this update of difficulty happens only once every 2000 blocks (2 weeks if we have, exactly, 10 minute blocks).  In our discussion here, we are considering short time spans, with constant difficulty.

It means that the hashrate of the whole network is such that on average, the whole network wins a good hash every 10 minutes.  If you have only, say, 10% of that hash rate, you will only win a block, on average, every 100 minutes.  As the "winning of a block" is entirely random, the "moments of winning" by a given pool are rarely coincident.  So MOST OF THE TIME, you win the block of which you find a good hash, because exactly at that moment, no-one else is winning a block.
It is only in those rare circumstances that you won a block and someone else did too, on the same old block, that one of them has to orphan, as decided by the other miners.

You cannot even use the time stamps at seconds precision, because the lower bits of the time stamp are used as nonce.

Most of the orphaned blocks are in reality much closer in time - simply because if the window of "collision" were bigger, there would be much more of them.

What you do, is post selection bias, however.   ALL orphaned blocks will be close in time !  Otherwise, they wouldn't collide !  But that doesn't tell you anything about all the non-colliding blocks !

If ever franky1's view were right, we would have (N-1) orphans about every 10 minutes, where N is the number of competing pools.  After all, according to him, there were 10 runners, and only one won.  So the other 9 lost.  Each 10 minutes, we would have 9 orphans.
In reality, we have 2 orphans or so per week.

As I outlined elsewhere in this thread, orphaning comes about because of the network delays between mining pools, when a pool didn't see yet that a new block was won, continued mining on the old block, and happened to win exactly during that interval, that block - which will be rejected by the other miners because they had already received the new block.

The probability of that happening is equal to (window of network delay) / 10 minutes in a Poisson distribution with average time 10 minutes and small delta-t.

From the orphaning rate, one can estimate the average network delay between miners.  It is essentially given by 10 minutes, times the ratio of orphaned blocks over the number of blocks won in a given period, because "around each won block" there is this "window of orphaning" which is about the propagation delay (including checking of validity).

Roughly, if we have, say, 2 blocks orphaned per week, and in a week, about 1000 blocks are found, we have:

10 minutes * 2 / 1000 = 1.2 seconds.

From this, I can also conclude that miner pools are DIRECTLY connected, because they can hardly receive their blocks through random paths in the P2P network where each node checks the block as a whole, in such small times.

These are also the reasons why I run a full node, on a old core-duo PC in my basement, which I upgraded with a big hard disk, and at which I look once a month to see if it is still running.  The other reason is to study the system itself.  I actually don't use my full node as a wallet: I use a light wallet on another system, and I connect to my "empty" full node in those rare cases I use bitcoin to pay for something (once or twice a year, say).

If you consider that such is even possible, that the "board of governors" of Bitcoin Inc sits together in a high level meeting to decide on the next laws of bitcoin (its protocol) - like they did on litecoin - then bitcoin is a centralized entity, I hope you realize that.

Next time, they might sit together on how to give themselves 10% of new bitcoin stash, after all, why not ?
A single extra coinbase in bloc XXXXXX.  Like all other "boards of directors" assign themselves some favors.

This.  It was designed that way.  I call that a "greater fool game" and as you say, if we run out of greater fools, we run to another game.
(but there's still a world of greater fools to take, so bitcoin is by far not "used up" as a game).

I had this argument recently with someone else on this forum: I think there is a lot of indication why centralized systems which "are easier to use" are largely preferred over decentralized systems, even if those last ones give you a higher sense of freedom.  This is one of the reasons why I think that decentralized crypto currencies will never take over the fiat system, or even come close.  (another reason is of course that most crypto has fallen in a speculative deflationary spiral).

When we look at several decentralized systems that existed before, and that have fallen out of favour, we find:
1)  the internet structure
The internet had a largely decentralized structure in the beginning.  In fact, it was its main invention !  The internet was invented as a military protocol to not have a single point of failure, let us not forget that.  The inventors of the internet had a decentralized P2P mesh structure in mind, so that when you bomb 2/3 of the network, the remaining 1/3 is still functioning.    Well, we see that, even though there is still some heritage of that, economies of scale have driven the structure of a mesh in rather wheel-and-spokes hierarchies, and very often, for your internet access, you rely on just a few commercial options.  You don't connect with a wire to your neighbours, who connect to their neighbours who ....
The internet is not entirely centralized either, there is no single boss of the internet, but there are main points of failure, where law enforcement can impose its rules and "look at everything", which wouldn't be the case in a true peer-to-peer network as it was intended.  It is not a mesh.

2) internet social content
the internet allows you to set up your own little web server at home.  You are entirely the master of that content.  You can run your own blog, and nobody can censor you (apart from the above centralization, which can censor your IP).  Almost nobody does (apart from people having content on Tor hidden servers).  People use centralized providers of these services, where they take an account and pay a modest sum for the service provided, or accept their personal information to be commercialized to get free access.

3) using centralized services like facebook.   Everything you do on facebook, you could do it with a home-installed blog.  The whole idea of the internet was to allow people to link content ; you could give restricted access to different users.  People prefer a centralized big company to do that for them.

4) discussion.  Usenet was a decentralized discussion system, that was entirely censor-resistant.  People dropped it to go to centralized discussion forums, because at least, there was a central boss (the admin) and his police (moderators) that kept rule and order.

5) storage.  You can buy huge amounts of storage for little money.  People put their stuff on the centralized "cloud" for easiness of access, and not to be bothered.

In all these categories, there are minorities sticking to their freedom and doing the decentralized thing.  But they are not the mainstream.

So, for payment systems, it is similar.  It is even worse, because trustless decentralized payment systems have an *increasing burden* on users as the network grows.  In fact, this is somewhat similar to usenet, which also had an increasing burden when its network became large (and ended up crumbling under it, with daily volumes of messages of several bitcoin block chains a day). 

A centralized system that works sufficiently well to allow people to do their thing, even if they are moderately badly exploited, seems in general to be preferred by the mainstream public, rather than the little bit of hassle of doing a decentralized thing yourself and taking responsibility for it.

It seemed unrealistic.

Care to explain ?  Suppose Alice and Bob set up a channel.  Alice puts in 1 BTC, and Bob puts in 1 BTC too.
The channel derives its security from the fact that the balance of the channel will always be between (2 BTC for Alice / 0 BTC for Bob) and (0 BTC for Alice, 2 BTC for Bob), right ?  It is a stack of non-published transactions from Alice or from Bob towards the "actual state", with "guard transactions" that allow the duped party to get everything (here, 2 BTC) if ever the counter party publishes a previous transaction that is not the last balance.  At least, that's my understanding of it.

Now, suppose that the network is such, that 5 times in a row, LN channels go through the Alice/Bob link in the same direction, from Alice to Bob, with 0.2 BTC to transact.  Now, Alice is at 0 BTC in the channel, and Bob is at 2 BTC in the channel.  There is no channelling possible any more from Alice to Bob.  Note that this probably means, that Alice had another channel open, say, to Joe.   If that channel was 1 BTC/1BTC initially, it is now also 0 BTC / 2 BTC, in the favour of Alice.

Alice still has her 2 BTC from the start (of which she locked in 1 BTC to Bob, and 1 BTC to Joe).  But it is now totally in her Joe channel, and nothing any more in her Bob channel.

You are saying that Alice doesn't have to settle.  She can wait for a payment in the other direction.  Sure.  But if her node is principally on a flux "from Joe to Bob" (say, Joe is closer to a big spender, and Bob is closer to a bitcoin accepting shop), her node is essentially "dead", and our 3 people have bitcoins locked up in channels that will wait for a long time before acting again.

The only thing to do, is essentially, to settle the channels, free up the 2 BTC that Alice had locked up in the channels, and start over again (after a day or so, when the time lock expires).

Did I get something wrong ?

This is probably a good reason to run a full node: to get a feeling of satisfaction of contributing "something" to this Great Network and sleep with the comforting thought of a deed well done.

Hahaha, the logical consequence of that is that the only way to ever get something out of bitcoin, is when you over-invested and were forced to sell !  Otherwise, you will only sell bitcoin when it is $0 !

There's really no point any more in trying to explain how mining works to you.

10 pools with each 10% of the hash power find, each individually, a single solution on average every 100 minutes (1 hour and 20 minutes).  On AVERAGE with an exponential distribution: so sometimes after 1 minute, sometimes after 5 hours.  Note that statistically, there is not the slightest influence *on what block* they mine.  They could change block at each hash, that wouldn't change their rate of success nor the statistical distribution of their finding solutions.  As such, they don't care when they have to switch blocks on which they mine: it doesn't change anything to their success.

Each time they find a solution, they win a block.  The chronological order in which the different blocs are found, is the block chain.  

Very rarely, it can happen that two pools find a solution within a few seconds interval, while they were mining on the same block - then, one of them will orphan.  The only reason why the second pool has an orphaned block, was that he wasn't aware he had to switch the block on which he was mining, and continued to mine on the old block (in vain), and happened to find a solution right at that moment.

If miners would know each-others' solutions instantaneously (no propagation delay, no verification delay) there wouldn't be any orphaning, because at the microsecond when a pool found a block, all others are aware of this, and switch to the next one.  So if 3 seconds later, a miner finds another block, it is already the following block, not an orphan on the old block.  Orphaning only happens because of delays between miners.

It is simply fascinatingly amazing to be talking about bitcoin technical aspects, and to miss this ultra elementary aspect.

This is a sentiment I'm having since quite a while, but for different reasons.  It is inherent in its design.  The small bloc is indeed a problem, but it is part of its "frozen" design.  My take on crypto currencies is now this: there are two kinds of crypto currencies: there are bitcoin-like, which are "immutable", frozen for ever ; and there are centralized developer coins, like ETH, DASH, Monero etc.... who have the habit of hard forking and evolving, but are also centralized with god-like dev powers that can do anything, any time.

Almost all of them have terrible design flaws ; but those that have centralized evolution, can try to improve ; the others are what they are, for ever, with their flaws.  The principal flaw of almost all crypto is the terribly deflationary nature of them (ironically, it is boasted as their monetary superiority), which turns them into speculative assets, and will never allow them to become reliable main stream currencies.  That said, them being highly volatile, but often also quite high-priced assets during their "time of adoption" (that is, when greater fools are still flowing in), they can be USED to transfer value and act as a kind of "payment channel", on the condition of getting in and out again very quickly (due to volatility, which would turn a payment otherwise in a speculative operation).

When using crypto as payment system, actually, the coin you use is in fact of no importance, from the moment that the customer can get them easily on exchanges, you can easily convert them back, and the transaction happens quickly, reliably and at low cost.   There can be extra requirements, and my own take is that the main reason to go through the hassle of using crypto for payments, is that it is a payment that cannot easily be done with the normal fiat system.  So one might like to use an obfuscated chain like monero, or ZEC.  (I wouldn't trust DASH too much for that, even though it is still better than bitcoin or litecoin in that respect).  If you're not into crypto speculation, on the side of the customer, or on the side of the merchant, there's no reason to keep the coins for any longer than needed (volatility risk), so the longevity of the coin doesn't really matter, it only serves for this particular payment.

Of course, you might want to use a crypto that does have *some* reliability and *some* volume, because of the hassle of implementing the payment system, you can't change the portfolio of currencies you accept for payment every 5 minutes.  So using crypto number 500 on coinmarketcap is maybe not a good idea, but if you hit in the first 50 ones, that can do the thing.

Using crypto as a long term "store of value" doesn't really exist in my mind: only long term hodler speculators are in this for a long time.  They don't want to "hold" their value, they want to become "immensely rich".

The nasty thing with crypto "immutability" is of course that it cannot really evolve ; so in as much as design errors are frozen in, and tech gets old, the only thing to do is leave the chain for a more modern one. 

As I said, and tried (in vain, apparently) to explain, you're totally wrong about how mining happens.  You think it is a cumulative calculation effort (even if you say you know it isn't, you think it takes very close to 10 minutes for every miner each time, whether that miner has 1% of the hash rate, or 40% of the hash rate, it always takes, up to a few seconds, 10 minutes). Can't help you any more.  This invalidates much of what you think about many things in bitcoin.

You're digging yourself deeper in the hole.  Of course, that is EXACTLY the reason why he only has a block every hour, and hence, about 1/6 of the total amount of blocks.  

This is the big problem, and why the LN has a HUGE centralization pressure.  Only hubs that can link to *a lot of customers* and hence lock in a lot of coins, and have *big amounts* locked in with other big hubs, can hope to average out a lot of payments so as to avoid channels to get exhausted.  If you only have, say, 4 BTC, and you lock them up in 8 channels, each of them 0.5 BTC, it will be hard for you, as a small node, not to "run out of BTC" in one of the channels after even just a few transactions.  If you have 10 000 BTC, and lock up 500 BTC in 20 channels to some other dolphins of the same kind, you will have a network that can handle a lot more transactions before having to settle, which puts you at a big advantage over smaller fish.  These lots of transactions will smooth out much more and keep your channel actually quadratically long alive with its content.  (if you put 10 times more coins in a channel, it will live 100 times longer, because the fluctuations go as square root of the amount).

It is funny that people say that, because I read that the Japanese government said something about *crypto currencies* and didn't single out bitcoin.

In fact, most jurisdictions will have to decide what asset class crypto currencies are in:
- illegal ones
- commodities
- currencies

or simply not say anything about it.

I would think that the statute of "currency" is not all rosy, because usually, handling a currency publicly requires stricter (banking-style) regulation than being a salesman of commodities.  On the other hand, usually currency acquisition is less taxed than commodity acquisition.