Showing cards when all in (and before decided how many times to run it) do you show?

Complete information to calculate the math yes. But if you're outs are in the muck, where does maths leave you?

With exactly the same probability. Look at it this way - suppose you have one out on the river. You know the identity of six cards, your hand plus the four on the board. Suppose your out is the eight of clubs. What is the probability that the eight of clubs will be the river card?

Suppose it’s a nine-handed table. Therefore there will be 18 cards dealt, followed by four more on the flop (burn plus 3 flop cards), two more turn cards (burn plus turn), followed by a burn card before the river is dealt. The river card is therefore the 26th card in the original deck. Dealing 25 cards before the river obviously didn’t change the identity of the river card. If we asked the question BEFORE dealing “what is the probability that the 26th card in the deck is the eight of clubs”, surely you would agree that it’s 1/52. Now suppose the dealer shows you six of the cards besides that 26th card, and none are the eight of clubs. You now have a 46 card deck so what is the probability that this card is the right of clubs? Does that probability change if the dealer moves those 25 cards from the top of the deck (including the six exposed ones) go another location before showing you card number 26?

Where you are getting confused is with information. The information we have does change our probability calculation. If we need that 8c on the river and someone says “Sorry, I folded the 8 of clubs”, then yes, we know calculate our probability to be zero, not 1/46. But that isn’t the actual situation in question. The 1/46 probability already accounts for the fact that the 8c might be a burn card, a card dealt to another player, or one of the cards remaining in the stub. Which of these is actually the case is irrelevant - none of them are the river card. It makes no difference where our card actually is; the calculation is whether it’s the river card or not the river card.

With the right decision.

This is just the same question as asking “if my quad outs are the next card to be dealt, where does the math leave me?” The card being in the muck or the next card to be dealt are in the probability calculation. That’s what makes it a probability calculation.

You are not offended by math, you just take it as a deeply personal affront.

You can take as little or as much from my posts, it's your choice and thankfully this is a place where opinions can be freely offered. You can also judge me, pigeonhole me and offer your insights into my psychological make-up as you wish, I'm really not offended by some random posts on an internet forum.

All I was trying to inject into the conversation is this -

Math is a human construct, developed to try and make sense of the chaotic world that surrounds us.

My opinion is, and this is unchanged, that it is folly to rely on only Math because sometimes, just sometimes, it will give you false hope in poker. Math is undeniably a major part of the game, but for me it is only a part of my overall approach.

I wonder why you think math gives you “false hope”. Math gives you precisely the exact amount of hope you should have in a given situation, and it actually quantifies that “hope”. In the one out case I discussed before, you will hit your out and win — SOMETIMES, but very rarely. More precisely if you played that spot some large number of times, N, you would win something close to N/46 times out N. Math can even do more than this - if you played this spot 46 million times, you would win somewhere close to 1 million times. But how close? Math can give us an idea how much of a range to expect, and it’s probably narrower than you’d think. In this specific case, there is a 99% chance that we would win between 997033 and 1,002,967 times.

Sure, math canÂ’t tell you if you will win THIS pot. But neither can your intuition, feel, instinct, or whatever you call it. Math can tell us whether or not we are making a winning play if we know our equity. That is the best we can do regardless of whether we use math or whatever else you think is relevant. Winning poker consists entirely of accepting risks in situations where the reward is sufficient to make the risk profitable and avoiding risks where the reward is insufficient to be favorable. If you can do the first more so than the second, you will win. If not, you wonÂ’t.

ItÂ’s just math. DonÂ’t believe me? Ho to a casino. Do you think they care what number comes up on the roulette wheel or what dice throw a craps player gets? They canÂ’t predict these outcomes any more than we can predict whether youÂ’ll hit your one-outer on the river. They make their money by taking risks at favorable odds and never taking risks at unfavorable ones. We canÂ’t totally duplicate this at the poker table; unlike the casino we will inevitably find ourselves in unfavorable spots. But the idea is the same - winning players will take favorable risks more often than unfavorable ones. It is math that tells us which are which.

This is a somewhat metaphysical question I suppose. It is possible that math is human created, or "invented" rather than "discovered".

Having said that, of all the things that humans think about across all of space and time, math is probably the one that has the highest chance of being universal across the universe, i.e. of precisely NOT being a human construct. (Physics and physical constants are maybe up there as well.)

That is the reason that the Voyager space probe's Golden Record starts with a depiction of numbers and the basics of addition and multiplication and fractions.

Of course.

It is in my view, something that has evolved from human's incessant need to understand through the creation of order.

You are most likely right, however there is always the slim chance that during the 'Evolvement' of the Math we have today, our human interruption sent us along a wrong branch which science has not been able to recognise and continues to build on what we have already deemed fact.

Numbers, Addition, Multiplication, Fractions....All human constructs.

Did this evolution of math begin before or after a game was invented with 4 sets of 13 cards with hand rankings with values reflecting their decreasing probabilities?

There's math for that!

You do understand that there are many variables that can only have estimates, right?

Disingenuous, but I'll play. Before. You used the word values.

YouÂ’re either being obtuse (perhaps intentionally) or the context was unclear. I was alluding specifically to the example of the river one outer. Math canÂ’t tell you if you will hit that out or not, but neither can your feel, intuition or whatever you are calling it. I have no doubt that once all the cards are dealt you can tell whether your opponents are weak or strong, but that wasnÂ’t what we were discussing.

Even with that, math can tell us how likely it is that a human will have a strong hand given some input about his range and how the betting action on prior streets went. For most players, this is more significant than any tells you get by the time the river rolls around, and IÂ’d suggest that you probably are incorporating such information into your reads as well. ItÂ’s certainly easier to think a villain is weak if he displays certain mannerisms and has check/called the flop, then gone check/check on the turn, followed by him checking to us on the river than it would be to make that read in isolation without considering how the hand played out. Math can lead us to very similar conclusions. Most players will take an aggressive action somewhere along the line with a strong hand. The absence of such actions leads us to assign a rather weak range with high probability, with a fairly low probability of an extremely strong holding that is trapping. Math leads us to the very same place in a spot like that as your intuition - mostly we are winning, with a very small chance that they are trapping.

I’d suggest that in general much of what we call “feel” or “intuition” is really just math in disguise. Many times that math is too complex to actually do in real time, but that doesn’t mean that the underlying phenomenon is not mathematical in nature. A baseball fielder (or cricket fielder if you’re not American) doesn’t consciously use math to catch a ball hit in the air. But it is nonetheless true that a robot that had good enough sensors to accurately measure the speed and direction of the ball off the bat, the ambient temperature, humidity, and barometric pressure, wind speed and direction and maybe some other factors I missed could in theory simply calculate the trajectory of the ball and move directly to the correct landing spot.

By analogy, even if we canÂ’t do all the math in poker, it doesnÂ’t mean that the math is not there and canÂ’t be used. The solvers indeed play a very good game and could likely beat most human players if they had sufficient processing capability to produce real-time outputs. Such solvers obviously would not rely on intuition or feel, but would solely be mathematical.

ROFL. Go on then, I'm usually keen to be educated.

My apologies, I wasn't trying to be obtuse. I skipped an important part of your reply that contained the context. Then I jumped on your first sentence in the 2nd paragraph (without the context) and it triggered me.

Thank you for being so persistent.

I understand what you are saying, but believe that we fundamentally disagree that intuition differs from a rule based principle such at Math. Despite anything else, how can Math possibly include all of the visual signals we get in a hand that helps us draw a conclusion?

I think that the heuristics argument is a simple solution that conveniently results in Math being the prevalent determine factor.

My sincere apologies for not reading your full post

I am surmising that using your line of thinking that we use history and probability within our brains to determine outcome which I can understand, but believe there is something else at play. Following your thinking, we could use that premise to describe the outcome of anything that happens in this world and that's a bitter pill for me to swallow and accept.

Guess we'll never figure it out then. Clear as mud

I wonder if there’s a way to calculate the percentage of the time the math gives you a false hope.

Yes, but it might give you false hope that you could know how often math gives you false hope. There’s probably math that tells you how often you get false hope about the false hope, but we’re getting into infinite regress territory here 😀

Always best to see what yer up against

We have a new rule, implemented by me, and it's a compromise. Now we just show ONE card before deciding how many times to run it. This keeps the maths right with EV but also gives you a decision on how many outs you might be up against.

Solved

A half-wit solution.

A solution in search of a problem.

backstairs, good for you that you seem to have the mental resources to devote to completely irrelevant poker questions. I guess you must have all the real poker strategy locked down.

Equivalent solutions:
1 Always run once
2 Always run twice
3 run once if the first card you look at is red, twice if black
4 opposite of 3
5 run twice if a National League team won the most recent World Series
6 run twice if the moon is full
7 run twice if Venus is in retrograde

Hopefully you get the point - it simply doesn’t matter how you decide - running it once or twice is completely equivalent in terms of win rate