Obviously, this can vary based on your position, the format of the game, the nature of other players and any action so far. But let's think about in general terms first.

Suppose you are dealt one hole card, and then have a chance to either draw a second card or shuffle this card back into the deck to draw two more. Where's your cutoff point for drawing one versus drawing two?

Presumably, the cutoff is higher for playing from earlier positions (and/or facing a raise) than from late position/heads-up. Maybe you'd fold Qx from UTG (full ring), but keep Jx on the button.

This could be answered empirically by filtering on PokerTracker. (Unfortunately, I haven't used it for years and don't have any data available myself). What do your numbers say?

The answer is more of a theoretical curiosity than anything. (However, there may be some practical benefit in PokerStars' "Power Up" format, where the "Reload" card allows you to discard one or both of your hole cards).

This is a curious indeed, as the playability of a hand obviously depends upon both your holecards. (65s and 44 are generally more playable than K7o, despite the latter having two higher cards).
50% of two-card hands are better than a random holding (with Q5o being the 'worst' hand with >50% equity vs ATC), but very few hands that don't contain an ace are actually profitable in the long run in 6-max, assuming 100bb deep. If the stacks are much shorter, then high cards become even more powerful. (e.g. With a 5bb stack, 43s lacks playability, but K7o would be a strong holding).

You could calculate your chance of getting a better holecard if you discard the initial one (e.g. "If I discard a nine, what is the probability I'll get a Broadway card?"), but that still doesn't solve the playability problem, as the majority of hands containing a random king, queen, jack or ten are still not +EV in most 6-max spots.
For reference, if a player wanted to use a 6-max UTG range of 66+,A2s+,K9s+,QTs+,JTs,ATo+,KQo (14.3%, 190 combos), over half of that range (102 combos) contains an ace. An ace is obviouslty a great card, so you wouldn't discard it! If the player's first card was a king, however, then there are only 50 combos in that 14% 'desired' range containing kings (out of all 198 combos that contain a king), so even with the first card being a king, it's quite unlikely that the UTG player will get a second card that makes the hand playable.

The situation would be much different on the button, however, as a decent proportion (maybe 35%) of all Qx and Jx combos are playable there, and close to half of all Kx.

Let me make sure I understand the question.

Given an established range for a position in NLHE, what is the rank of the median hand such that a random hand is equally likely to be better or worse? Further, use this to know which rank of a first card is worth keeping and only drawing one more card versus discarding and drawing two.


Take your range from a given seat and write out all the combos. Be sure to account for more combos of unpaired hands than paired hands.

Rank all of these in order best to worst. You can use raw equity or post flop playability according to your preference.

Find the median. This is the geometric center of your weighted range.

Now, the median of the deck is Q8o, iirc.

However, a UTG range might consist of 85 percent combos above Q8o, and 15 percent combos below Q8o, just for example.

Once you know the geometric tipping point of your range, you will know how to play the poker variant. So if your first card dealt was an 8, then you probably need another 8 to get a playable hand UTG. There are 3 cards in the deck that are playable, versus just discarding and drawing two more. On the other hand, if your first card is an Ace, then you may decide to play any A, K or Q, plus any suited J down to 5. If so, then that is at least 17 cards in the deck that are favorable to draw.

This process goes for every seat, all the way to BB.

Lets say BTN raises 3x, sb folds, and you are BB. You decide to defend the top half of your range. If Q8o is the median, then you defend any pocket pair, any A or K, and Q8+, plus T9s, J9s and JTs.

Quick double check on Pkrcruncher shows that we are only defending 39.4 percent of raw equity. So we are missing some suited connectors that are equal or better than Q8o given a full board of Holdem. Plus we have some combos that are above Q8o in raw equity but fare worse than Q8o across a full runout of holdem.

Pkrcruncher would defend this range of 49.8 percent combos:



However, given the discard option variant of poker where you get one hole card and can discard it or keep it, the suited combos from Pkrcruncher will be hard to draw to, given that there may be sometimes only a few cards in the deck that would make a playable hand, given the first card is a 6 for instance.

So, we go back to our situation in the BB. We need to know the rank of the first card so that we decide to keep it instead of discarding it. That seems to be the question at hand.

We know we are keeping any A or K, probably any Q, so lets investigate getting dealt a J.

If we are dealt a Jack, then we are happy with any A, K, Q, J, T, 9, 8, 7, and will accept suited 6 to 4. That is 33 cards in the deck of 51 remaining that we can draw. So we keep the Jack when we are dealt a Jack.

Next up is a Ten. If we are dealt a Ten, then there we will be happy with any A, K, Q, J, T, 9, 8, and suited 6 or 7. That is 28 cards in the deck of 51.

Now, with the above method, the playable hands with a 9 or 8 dealt first will deminish quickly. Also, it is more complicated than this. If you are dealt a 9, then you are drawing for the lower side of your range versus taking a chance at discarding and redrawing AK, for instance.

This is where I bow out and leave the fun for you and other posters to have fun with it.

I'm pretty sure that if I'm UTG, I want 100% of my range to be ahead of Q8o. But I'm a nit.

It seems to me that anyone with a significant PokerTracker database could find a good answer by filtering for all hands sharing a given card (and could further break this down by position) and comparing their winrate with those hands at that position with their overall winrate at that position.

(Though, I think you need to double the weighting on the corresponding pocket pair. For instance, Qx has three "outs" to make QQ and four to make QJ, but there are six overall QQ combos versus 16 QJ ones. This is because half the QJ combos are dealt J then Q).

You would need an absurdly large database for this to be accurate. Also, the EV's of one individual is going to be different than another's due to differences in strategy. Consider the fact that someone might open 98s UTG, and another may just fold it. This will skew EVs as well.

Sure, getting exact answers will be difficult, but there's such a sizeable jump between e.g the set of all Qx hands and the set of all Kx hands that it should be apparent where the cutoff is for relatively modest sample sizes.

Differences in strategy shouldn't matter too much unless someone is playing really suboptimally, because close decisions should be ~0EV anyway.

One tractable way of approaching this, though far from ideal, is to derive the optimal "switch" strategy which maximizes your chances of having a starting hand you would open given your table position.

This can be done utilizing opening ranges of every position at a full-ring or 6-max table, whichever is appropriate. Opening ranges have become fairly standardized over the years though, of course, there is still quite a bit of variation across players.

Since a recent thread has found that different ranks are folded at significantly different rates, frequencies of folded cards of players in front of you can also be taken into account in considering the probability distribution of the "new" card you would be dealt if you decide to switch. (These can be determined fairly accurately via simulation as in the previous thread.) Accordingly, performing this analysis seems pretty straightforward in the scenarios in which it is folded to you.

Theoretically the previous simulations could be extended to derive as many and as much granularity in the resulting probability distributions as you desire if one or more players has entered the pot in front of you.

Of course, the major downside of this approach is that it does not take into account the amount of chips you are likely to win (or lose) when opening each possible hand in your opening range. Presumably, there are no hands in your opening range that have negative EV?

There's no sizable (or even discernible) jump in the EV between (playable) Kx and Qx hands in modest sample sizes. At the top of a profit list in most HEM databases will be AA (by a long way), then there's KK and a big gap to QQ, then it's pretty close between JJ, AKs, AKo, AQs, TT, AQo (or a slightly different order). And then there's the rest.
In a human lifetime of play at 6-max, it's impossible to even prove if KTs is better than JTs UTG. Both have EVs that will be very close to breakeven, and so are therefore subject to massive amounts of variance.
Anyone with HEM can look up their results and find absurdities like being profitable with A3s while losing with ATs or AJ, or crushing with 77 while getting destroyed with 88.

FWIW, you can look at the old EV chart from a sample of 115 million hands of fixed limit at https://www.tightpoker.com/poker_hands.html and note how few hands were +EV, and that many were very close to zero. One could use those results to say "98s, T8s, K7s, and A2s are all equally good, and are better than 87s and Q8s, but not as good as K8s" but an individual's skill level (and luck) could produce completely different results.

To go back to the original question, I think I'd discard everything except an ace, except maybe on the button, where kings might be worth keeping. Overall, the majority of Kx hands haven't made money for me. I would be discarding my first card very often in this game. A blank K is no use to me when I could fold it and get 65s instead!

Just looking around google I found this:
https://www.tightpoker.com/hands/ev_position.html

If we find the weighted average of all the profitable hands on that page, we get that UTG makes about 4.9BB/100 hands. If we find the weighted average given the first card is a king, we get 11.7BB/100 hands. Clearly we should not discard a king UTG if these were our own stats. We should not even discard a jack. The page doesn't show the profitability of all Jx hands, but holding a jack just to hope for JJ has an EV of 4.9BB/100.

Haha, I just realized Arty posted the same link above me. I hadn't read his whole post.

I don't see suited connectors?

Interesting chart, thanks. I'm a little surprised that you'd keep Jx, but I guess it makes sense that the equity of high pocket pairs make it worthwhile.

Also interesting that the biggest losers in this database are not the absolute worst hands like 72o, but some weak suited hands and Ace-rag. I suppose it's because even beginners know to fold 72o, but they might be tempted to play some "pretty" hands in decidedly -EV situations.

This seems wrong for the reasons browni listed. Sure, the majority of Kx hands aren't profitable, but the majority of random hands aren't profitable either. Kx has a ~14% chance of turning into the solidly profitable hands KK or AK, whereas random cards have a much lower probability of turning into a premium hand (~2.5% for QQ+, AK). For what you're saying to be true, the overwhelming majority of your winnings would have to come from AA and that's not the case.

Another thing is that the answers to the question "What's the breakeven rank of one hole card in real life NLHE?" and "How should you play a hypothetical NLHE variant where each player gets to see their first hole card and has the chance to replace it?" are quite different. I began the thread with the former question in mind, but the latter is somewhat interesting as its own separate question.

For instance, if all players can see if you replace your card, and you only ever draw 1 with a high card, then you could never represent anything on low connected flops. So for deceptive purposes, you'd need to occasionally draw 1 with a low card in order to be able to get suited connectors in your range.

There could also be notable card removal effects. For instance, if a number of early position players draw 1 then fold, that tells us more about the composition of the deck than it would in regular Hold'em.

I had a rethink about the strategy after posting and realised I was being way too nitty, due to my horrible understanding of basic probability. Hopefully my schoolboy maths will fare better in this reply, or I'll have to give up poker.

If we completely ignore the EV of specific hands and just look for playability (e.g. we consider AA and JTs to be equally playable, even though they are, in fact, massively different in value), then we can formulate a discard strategy based on the probability of getting a playable hand from a 'standard' range, as touched upon by whosnext in post #8.

e.g. Suppose hero is UTG at 6-max, and his arbitrary/standard playable range is 66+,A2s+,K9s+,QTs+,J9s+,T9s,AJo+,KJo+ (just under 15%).



Obviously, the chance of hero being dealt a playable hand in that seat is 14.93%, so after looking at the first card, hero should calculate the probability of getting another "useful" card if he keeps it, and compare that with the probability of getting a playable hand if he discards the first one and takes two at random (which we already know is 15%).

All aces are obviously kept, since almost 50% of the desired range contains an ace, so you're halfway to having something playable when your first card is an ace. A nearly 50% chance of having a playable hand (instead of the 15% chance with two random cards) makes an ace a clear keeper.

For other cards, I arbitrarily decided to use hearts as the suit of the first card, in order to make counting playable combos easier.

If hero is dealt the king of hearts, what is the probability that the second card will make his holecards playable? His "outs", as it were, are 4 aces (to make AK), 3 kings (KK), 4 queens (KQo/KQs), 4 jacks (KJo/KJs), plus the ten of hearts (KThh) and nine of hearts (K9hh). These outs total 4+3+4+4+1+1 = 17 out of the 51 cards left in the deck. In short, when hero is dealt a random king in this spot, he has a 17/51 = 33.33% chance of his second card making the hand playable.
Since this 33.33% figure is higher than the 15% chance of getting a playable hand from two random cards, the strategy with a king UTG would be to keep it, not discard.

How about Qh? The outs are 4 aces (AQo/AQs), 4 kings (KQo/KQs), 3 queens (QQ), plus Jh, Th (QJhh, QThh) = 13/51 = 25%, so a queen should also be kept, not discarded.

Jh? 4 aces, 4 kings, 3 jacks, Qh, Th, 9h = 14/51 = 27% => keep a jack too.

Th? There were no offsuit combos containing a ten in the arbitrary range, so the outs are 3 tens, Ah, Kh, Qh, Jh, 9h = 8/51 = 15.7%.
Amazingly, when dealt the ten of hearts UTG, we'd still be better off keeping it (15.7% chance of getting a playable hand) instead of discarding and having two at random (just under 15%).
The 9h would have 3 nines, Ah, Kh, Jh, Th = 7 outs, so this would be discarded as 7/51 = 13.7%.

You could do similar simple probabilistic calcs with the wider ranges for other positions. It would be quite interesting to look at, say, a 40% range for the BTN. Taking card removal effects at this juncture would be going a bit too far imo.

Note: I may have made some stupid combo-counting errors (it's late and I'm tired) but I think you get the point.

I might address this (the actual EV of specific hands) at a later date, because it's actually kinda frightening just how profitable AA is in comparison to other playable hands.

I know OP was not really asking about the power up variant strategy, but I want to add that the overall ranges of players ahead of you that have a playable hand should be stronger, and further tightening of your range and avoiding marginal hands multiway should be GTO.

I think you would actually discard 10h just because 10h does not give you the chance of having the top of your range, yes you will be able to vpip more often but you have no chance at getting JJ+ or ak

Well, yeah, that's kind of the argument I started with. If you want to make the most money on that single hand in a vacuum, then folding low ranked cards makes sense. Getting an ace as the first card (instead of a ten) means you have about a 5.8% chance of getting a hand (AA, about 10EVbb) that is roughly five times more profitable than TT (EV of about 2bb), and about a thousand times more profitable than T9s (EV of about 0.01bb).