My math star faded long ago but sure.
First an example. Let’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 “out there”, 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 it’s 31.25%).
For a single card that’s fine, but for the next TWO cards, you can’t just add the 31% again and say “ok so I’m 62%”. To understand why, let’s pretend we’re big money players on tv, we ship, and we “run it twice” - you’ve probably seen that happen. So, we’re going to do another two cards “twice”, thus we’ll see 4 cards total.
If I just add 31% for each draw, it seems like I’m OVER 100% to make my draw at least one time. That’s clearly impossible, right? With each draw, the odds of me seeing an “out” 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 we’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 I’m going to just consider the first two draws (which is what we see IRL anyway).
We’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 “HIT-MISS”) the odds must be MULTIPLIED, then the whole summed. We’ll use 30% as our “HIT” 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 we’d get if we just added the simplified 30% number. It’s a coin flip.
“OK GAMEY COOL STORY BUT HOW DO I USE THAT AT A TABLE GOOD LORD”. 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 generally…
- 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 “complement” of that 49%, so again, 100% - 49%
That’s your ballpark number. Gets hard for, like 8 outs. It isn’t a bad idea to know the basic flush (9 out) and straight draw (8 out) by heart (around 34% and 31% respectively). Basically you’re a 2:1 dog in those situations, which is why if you’re betting AGAINST those draws, you want to bet somewhat over 1/2 the pot, to “price out” the draws… 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 you’d think based on this loooong post. For that matter, I also believe in intuition(!), but that’s a very deep topic. I’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 “price” a call, esp when folks are throwing really small bets at big pots and I have only (say) 3 outs.