Just wondering if anyone can show me the mathematical proof of two over cards vs a pair pf equaling roughly a 50% chance of winning by showdown.

Related question...is there a way to calculate pf equity going into the flop? Since equity calculated pf assumes you go to showdown

I don't think there is a mathematical equation that will satisfy you. Typically the answer to such questions as, "what is my equity against this hand?" are calculated by brute force, full-enumeration where every possible outcome is examined and counted as either a win, a loss, or a tie.

For example, pokerstove just deals out all of the possible boards and counts up the number of times each hand wins, ties, or loses.

What does this quesiton mean?

"Equity" means what portion of showdowns you are expected to win. *

Obviously the cards only decide the outcome of the hand if there is a showdown. The cards don't decide anything if the hand ends on the flop or turn.



* technically it's what portion of the showdown pot but that only matters in multiway hands, heads-up it's exactly the same thing.

So...rephrased question

Is there a way of estimating our chance to be ahead on the flop vs a hand and/or range? For the turn and river, obviously we have the 4 and 2 rule.

I used to do scenarios like that with poker razor but I'm not sure if you can still get it.

It's considerably more complicated preflop, because usually when you are trying to figure out if you'll be ahead on the turn or the river, you're drawing to a really dominant hand (straight or a flush usually, or 2 pair, etc). Your opponent will usually not have that much of a chance of re-drawing or simultaneously outdrawing you, so even when you're drawing to, say, 2 pair, it's not that likely that he'll also make 2 pair or better to beat you.

But with preflop matchups, except in the case where you flop trips or 2 pair, it's not likely that you'll have nearly a sure winner, and it's possible for you to hit your outs and get outdrawn at the same time.

But sure, there are ways to figure out your chances of hitting single-card draws, that is, pairing one of your cards, so that if you have a non-pair hand, you'll make a pair, or if you have a pair, you'll make trips.

OK, so. You have 2 cards in your hand and 50 unseen cards. If you have 1 out, the chance that the first card on the flop will be that card is 1/50. The chance that it will NOT be your out is 49/50. If the first card isn't your card, the 2nd one has a 48/49 chance of not being your card, and the 3rd would have a 47/48 chance of not being your card.

So the odds of the flop not containing your 1 out is
49/50*48/49*47/48 which is 47/50
so you WILL hit one of them 3/50 times. Geez is that right, that seems too easy.

That implies a rule of 6 (which actually might be exact? I need to double check this) i.e. each out has a 6% chance of hitting the flop.

I've never done this calc before. Let me double check this and get back to you.

Obviously if you have 1 out with 1/50 chance per card and 3 cards, your chance of hitting it is 3/50.

OK, it looks like
1 out: 6%
2 outs: 11.76%
3 outs: 17.27%
4 outs: 22.55%
5 outs: 27.60%
6 outs: 32.43%
This is, of course, your chance of hitting (at least) one of your cards, it doesn't take into account what your opponent may hit.

It is obvious in retrospect, but only because it's 1 out. I sometimes have mental roadblocks. If you were using 2 outs, it's not so neat, because of the potential overlap, the same reason that your chance of hitting turn or river isn't just twice your odds of hitting the turn (although it's not far)

Edit: my problem here, I guess, is that I work with probability enough to be familiar with doing the calculations, but not often enough that I have an instant intuitive grasp. It's the same with most of calculus and stuff for me, I often have to do all the work to get to the end and see the obvious conclusion.

Thanks a lot Rusty, I really appreciate it.

The reason I was asking is because spots come up where I try deciding if it's worth it to flat with AK and if it isn't the worst thing in the world if I ck/f the flop if I miss and wait for better spots, instead of trying to call flop bets with 6 outs that might be dead. Obv these flop probabilities/strategies/etc aren't discussed in any book (that I know of, anyway), but it's just stuff that I'm thinking about on my own.

Since we're talking about showdown equity; you can just continue your computation over 5 cards;

49/50 * 48/49 * 47/48 * 46/47 * 45/46 = 0.9
1 - 0.9 = 0.1 or 10% chance of hitting your one-outer. As alluded to, though, this doesn't account for the fact that a card could hit to give your villain a better hand. In reality, I can't envision the pre-flop scenario where you are drawing to one-out, never mind the weird combinations where the board winds up giving you four flush cards or 4 straight cards.

Let's say it was AK vs. QQ...
So for these 6 outs, the calculation looks something like;
42/48*41/47*40/46*39/45*38/44 = 0.4968
1 - .4968 = .5032 or about 50%. But this is not counting the times where one or two of your overcards hits AND a queen comes, I can estimate that by considering, of the 48 unseen cards, the chances that one of your desired cards (6/48) AND a queen (2/48) hits is the product of their probability, which is about 0.52%. HOWEVER, this event could happen in 10 different combinations [5C2 = 5*4/2 = 10] (we won't use arrangements or permutations since it doesn't matter if the queen hits before or after your ace/king), so it's more accurate to say this event could happen 5.2% of the time, so if I subtract that from my first calculation, I'm now looking at a preflop equity of 45.12%, and I STILL haven't accounted for the possibilities of a straight coming in, a 4-flush hitting, a 3-flush hitting if my AK was SOOOTED, etc.
Anyway, we like to say "coin flip" on two overs to a lower pair, but, except for a pair of deuces, the pair is always ahead of you PF.
Ultimately, propokertools.com is your friend!

Thanks for clearing that up for me. It seemed that intuitively, half the deck should equal 50%...OBV I never held this speculation throughout my poker career, I just learned the math for coin flips, 4 and 2 rule, etc and accepted it. It's nice seeing the proof for it. Again, thank you.