Observations on 'Non-Random' Elements

OK, OK, I know the claims of non-randomness have been debunked.
BUT
There are some peculiarities which can be useful to know

• After the flop one of the three cards will often be duplicated for the river and turn - more often than you expect. This reduces the chances of getting a straight or flush. Yields more triples

• If the flop has three low cards, the river and flop will often also both be low cards

• It is uncanny how often a pair that you fold ends up scoring two pair or a full house!!!

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to many for me to list.

Do you have some exact numbers how often each of your stated events did happen and didnâ€™t happen?
Because there is someting called â€śconfirmation biasâ€ť. Almost every time one of your event happens you notice it. But if it doesnâ€™t happen you wonâ€™t recall it.

Even if you compare these events with the with the expected value of them, it can happen that these happen more frequently over a short run, than to expect. Over the long run however your expectation and the number of events will come closer.

There are several different types of memory errors , in which people may inaccurately recall details of events that did not occur, or they may simply misattribute the source of a memory. In other instances, imagination of a certain event can create confidence that such an event actually occurred. Causes of such memory errors may be due to certain cognitive factors, such as spreading activation or to physiological factors, including emotional factors.

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That shows even more, that we need actual numbers for these events.

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@JustlyFishy
How often do you think a paired board should happen?.
Assume the flop is 4s 5d 6c what are the odds of the board pairing on the turn or river?.
Simple maths.
There are 3 fours + 3 fives + 3 sixes (total of 9 cards) left in 49 unknown locations. Thatâ€™s 9 cards in 49 which is 18.36% on the turn + 9 of 48 on the river which is 18.75%. That totals to more than 37% or well more than a third of the time.

Exactly

@feelmysins Not quite how the math works. Turn is 9/49, River is 9/498/49 (pairing twice) + 40/49 (miss the turn) * 9/49 (pair on river instead). So (949 + 98 + 940)/(49*49) = 36.35%. 37% is an ok approximation since the single card odds are low.

@gameygamer
Went with the simple math version so it can be done on the fly at the table. Should your calculations be done as 9/49 on the turn and 9/48 or 8/48 (if the turn hits) on the river. The turn card is now known so only 48 cards remain in the deck. River card can only be either 9/48 or 8/48. I may be wrong and if I am I would really appreciate if you could elaborate and explain to a less than stellar mathematician.

My math star faded long ago but sure.

First an example. Letâ€™s say you have AKs, miss the flop but have 4 to a flush. You put your opponent on some sort of pair. You have 9 outs for the flush and 6 outs for the over cards or 15 outs total. There are 48 cards â€śout thereâ€ť, so on the first card, you have the plain 15/48, which is (and this is quick table math) a little over 30%, like 31% say (checking a calculator itâ€™s 31.25%).

For a single card thatâ€™s fine, but for the next TWO cards, you canâ€™t just add the 31% again and say â€śok so Iâ€™m 62%â€ť. To understand why, letâ€™s pretend weâ€™re big money players on tv, we ship, and we â€śrun it twiceâ€ť - youâ€™ve probably seen that happen. So, weâ€™re going to do another two cards â€śtwiceâ€ť, thus weâ€™ll see 4 cards total.

If I just add 31% for each draw, it seems like Iâ€™m OVER 100% to make my draw at least one time. Thatâ€™s clearly impossible, right? With each draw, the odds of me seeing an â€śoutâ€ť card should approach 100%, but never exceed it (if we drew down the whole deck and got to the last 15 cards having seen NONE of ours, the odds of the next - and each subsequent card - being one of our outs is 100%).

There are a couple of minor problems in our calcs - for instance, each draw is from a shrinking set of unknown cards (48, then 47, then 46, then 45), also I might pair up on one card but Villain gets two pair or trips on another. But those are not too significant and weâ€™ll ignore them for now.

The real flaw is that you have to consider the logical unions of each draw with all the other draws. The math is pretty complex, so Iâ€™m going to just consider the first two draws (which is what we see IRL anyway).

Weâ€™ll ignore all the trips and 2pair combos for our villain and just focus on our outs. With each draw, we either HIT or MISS. So between the two cards, we could HIT-MISS, MISS-HIT, HIT-HIT, or MISS-MISS. For each of those (like â€śHIT-MISSâ€ť) the odds must be MULTIPLIED, then the whole summed. Weâ€™ll use 30% as our â€śHITâ€ť to keep it simple, ignore the diminishing card pool and the various Villain spoilers.

• HIT 30% X MISS 70% = 21%
• MISS X HIT = 21%
• HIT X HIT = 9%
• MISS X MISS = 49%

21 + 21 + 9 + 49 = 100. The odds of us hitting at least once are 51% - not 60% which is what weâ€™d get if we just added the simplified 30% number. Itâ€™s a coin flip.

â€śOK GAMEY COOL STORY BUT HOW DO I USE THAT AT A TABLE GOOD LORDâ€ť. Firstly, the smaller probabilities CAN be safely estimated by just adding the % for each draw (so a one outer is around 4% or a little less). But generallyâ€¦

• count your outs (say 15)
• multiply by two and subtract that number from 100 (2x15 = 30, 100 - 30 = 70)
• multiply that number, as a %, by itself (MISS MISS), so 70% (.7) squared = .49 or 49%
• your odds are the â€ścomplementâ€ť of that 49%, so again, 100% - 49%

Thatâ€™s your ballpark number. Gets hard for, like 8 outs. It isnâ€™t a bad idea to know the basic flush (9 out) and straight draw (8 out) by heart (around 34% and 31% respectively). Basically youâ€™re a 2:1 dog in those situations, which is why if youâ€™re betting AGAINST those draws, you want to bet somewhat over 1/2 the pot, to â€śprice outâ€ť the drawsâ€¦ more math there lol.

Final caveat - I believe the most important thing at table is reading, not math, so I do a lot less of this than youâ€™d think based on this loooong post. For that matter, I also believe in intuition(!), but thatâ€™s a very deep topic. Iâ€™m just answering the question (I tutored math once upon a time), and probabilities can be tricky. I mainly use this sort of thing to â€śpriceâ€ť a call, esp when folks are throwing really small bets at big pots and I have only (say) 3 outs.

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In Hold-em, the flop will be double suited about 60% of the time. For it to become triple suited is not much of a stretch. The exact stats are in many poker books.