Hi guys,

This is Nate Silver of the FiveThirtyEight blog. I used to post around here a lot as Nate Tha' Great a few years ago when I was a limit donk.

I'm working on a book about forecasting and prediction called THE SIGNAL AND THE NOISE, which is due to be published this September. The book contains a full chapter on poker and the chapter contains a sample hand, which I will run by you here. I'm not super concerned about anything here (e.g. bet-sizing) until the all-in river decision. In the book, I assert that the river decision is quite clear, but I wanted to see if folks here share that impression.

Here's the hand.

You're playing a $5/$10 NLHE game at the Bellagio. It's folded to you in mid position, where you raise to $25 with 8 8. A lone opponent in the cutoff seat calls. He has $1360 in front of him before the hand begins and you have him covered. Your reputation is about average and you've had a decent session, mixing your play up a bit not getting too out of line.

The opponent is some sort of lawyer from the East Coast, whom you gather visits Vegas a few times a year to play golf and poker. He's in his 60s and has some money to blow, but he's been at the table for a few hours and you gather than his play is fairly ABC, leaning a bit toward passivity. He's not playing too many hands per se, but maybe more than he should since he's a fairly mediocre postflop player. You thought you saw him bluff once but not in a huge pot. He's not a total fish by any means but is probably a slightly losing player in this game, even before the rake.

The call (rather than 3-bet) is fairly common for him in position. It probably means pairs and a lot of suited stuff. You're not sure how often he'd play middling offsuit aces like ATo and offsuit Broadway cards but they're perhaps not totally foreign to him in this spot. With all that said, you didn't know him from Adam a few hours earlier so these reads are solid but not foolproof.

The flop is K 9 3. You bet $35 into the $65 pot and he calls fairly quickly.

You spike a set on the turn with 8. You bet $100 into the $135. He calls again without making a show of it.

The river is the 5. You bet $250 into the $335 pot. He says "all-in" fairly quietly and somewhat neatly shoves his remaining ~$1,200 in chips into the the pot.

You're calling $950 to win $1,770 so you need to have the best hand about 35 percent of the time to make it correct. How often do you think you're ahead? If you could give your best estimate (XX percent) of the probability given the information I've given you and your general poker knowledge, that would be cool -- this is not a trick question or anything.

thank you,
Nate

Given the description, it seems unlikely that villain would bluff here given the bet size on the river. Assuming that he only bluffs say with 10% of his flush draws that missed and that he is somewhat loose, I will assign him this range of club draws (AQs-A4s,A2s,Q7s+,J7s+,T7s+,96s+,86s+,75s+,64s+,54s). That is 32 combos, so if I assume he is bluffing 10% of the time with those hands, that is 3 card combos.

33 is 3 card combos and I would say that he would play those 50% of the time. K5s is 2 card combos that would surely play this way. 89 I guess would play this way 20% of the time (10 card combos) and I would give AK and AA the same 20% probability (12 card combos).

We lose to 99 and 67 of clubs. That is 4 card combos. I would say 99 has the same likelihood as 33 before in playing this way 50% of the time. We lose to 2.5 card combos. We win vs. the 10.9 card combos I gave him earlier. I would say we win here 81% of the time.

call because even if you are behind of a few hands, there are more hands he is value jamming the river with that you beat. if villain bluffs ever then it goes from a call to a celebration call

edit: prob whatever perdoom said

2+2 celebrity alert. you're the man, nate.

76 is gone on the flop unless it's specifically 7c6c, so the nuts is very unlikely on the river. KK 3-bets some percentage of the time preflop. his river-shoving range is probably something like KK, 99, 33, and the occasional K9, K5, K3, or 98. i think he turns a missed draw into a bluff very rarely.

it's a trivial call given the pot odds; i'm not quite as optimistic as perdoom ("I would say we win here 81% of the time"), but i estimate that we're comfortably ahead more than half the time.

i agree that we're likely to be quite far ahead of his range but i'm not sure i could be more specific without more information.

i do think perdoom's range is a little optimistic though, i think its not impossible he bluffs 0% here ( some players, especially recreational players, could go years without bluffshoving a river) and i think AK and AA are more unlikely than the 20% he assigns - its just too much of a parlay for him to have flatted preflop and on every street with them and now he decides to raise the river. not impossible but not very likely at all in my opinion.

i'm a big fan btw nate!

calling but not happy at all. Expect to see 67cc and 99 here a LOT.

Don't forget we need to further discount KK, 99, and 33 because they would raise the flop and turn sometimes. I'm never folding here because there's a reasonable chance he could be turning a missing flush draw into a bluff and could also have 33 as well as a weirdly played 98. He shows up here w/67cc a good amount of the time but that's just the price of doing business

50%-80% seems like a reasonable range, 99 and 33 most likely hands.

Without discounting anything at all we're 30% vs KK, 99, 67cc, 33. Folding is not an option. That's 10 combos, even if he's almost never on a random bluff/value shoving AK or whatever we're good at least 50-60% here.

I can't believe the responses I'm reading.

This looks like a fold, I'd say its decently close, 10%-20% was my answer.

These kinds of villains DO NOT think 33 and 99 are the same hand (correctly), his valuerange is probably perfect, or close to perfect. And his blufrange is next to nonexistent.

Have you guys ever played live before?

Also, I don't get the title, how is this a sanity check, the hand is really close therefore reasonable arguments can be made for any decision.


good handreading, the hard part is accepting that you should go home curious instead of broke

Ranges do not equal frequencies.

I would include almost all of 67cc, and I would discount half of the oversets (they play faster sometimes), and id discount much more than half of the undersets (they play faster sometimes, and they often calldown instead of raising river from described villian.)

Lol, huge range of answers in the poll (how someone can vote 0-10 or 90-100 I'm not sure). I agree that this is a clear call. I voted that we're good 60-70 % of the time, and I don't think that's too optimistic. Expect to lose to 99, 67cc sometimes sure, but folding seems ridiculous given he could definitely play 33 like this (I think discounting over 1/2 33 combos is too much), and it wouldn't surprise me at all to see a mediocre live player jamming some 2 pairs combos.

After playing alot of live in thepast year, coming from 60+ year old passive guy , this is going to be 67cc 95% of the time and kk 5% of the time .

I would fold face up

Come on now, OP described villain as a lawyer tourist type with lots of money, not some huge nit live pro. I don't think he's going to have a perfect value range. I won't play all KK/99 combos this way because of the call pre and he might raise some of the flop. I'd add a few combos of random Kings up and 33. Then add a couple combos of missed draws. Sure the one 67cc combo fits. I'd call but not happy about it. I voted 40-50%.

Also I'd raise bigger pre in a live game.

Obviously you need to discount certain hands, point is no matter what you do he's only going to need a few combos of random bluffs/worse for value to make this a profitable call.

If you have a strong enough read after a few hours to eliminate those combos from his range with 100% certainty you must be some sort of mind reader.

You don't need to be a mind reader to make an educated guess, and you don't need 100% certainty, you need 75% certainty, or whatever.

agreed, except only fold face down, usually

Discounting as you wanted to we're losing to 4 out of 5 combos. You only need 2 additional combos to make this profitable, yes you need to be a mind reader. 75% is not good enough.

Assigning him 2 combos of 99 and 33 each and 0.5 combos of KK after discounting from preflop and flop/turn actions (assuming he 3bets KK like 75% of the time and assuming he plays flop/turn differently with KK/99/33 at some point 1/3 of the time) whilst giving him 1 combo of 67cc, we are good 2 out of 5.5 times.

This is good enough for a sigh call, let alone the fact that we may assign 0.5-1 combos of random spazzes in his range either from a bluff or a thin value shove with 89/K9 or whatnot.

4 out of 5?

Okay I'll just go ahead and do it for myself.

67cc, 1
KK, 1
99, 1.5
33, 0.5
other stuff - X

I believe X is much lower than it needs to be for us to find a fold.

It doesn't take mind reading to make an educated guess, I could argue that it takes "mindreading" to lump other hands into his range, lol.

omfg this, this and this! what a bunch of nit tard newbs.

Glad that this one sparked some discussion. It's fascinating that, at least in the poll, we had people who thought we're almost always ahead and others who thought we were almost never ahead.

My view is that this hand is a pretty easy call.

The insight here is that I think people are narrowing the opponent's hand range a bit too much. There are a lot of low-probability combos he could have which collectively start to weigh pretty heavily against the chance he could have us beat.

What I actually did in working through this hand before I posted it was go through the conditional probability of every hand combination on every street, i.e. if he had exactly 55, how likely is it that he would (a) call preflop, (b) call the flop conditional upon his having called preflop; (c) call the turn conditional on his having called the flop, etc. Obviously this takes a LOT of time and requires you to make a LOT of assumptions. With that said, this process (essentially a matter of Bayesian updating) represents the Platonic ideal of how we should think about our opponents range. When I was playing poker actively, I never went through a hand in quite that much detail and I found it pretty interesting to do so. Anyway, here's about what I came up with:

Combos we lose to

Obviously, he could have 7 6 . From start to finish, that hand is the most consistent with his play. Fortunately, it's just 1 combination. It's almost impossible for him to have any other 76 combination since it did not make a straight draw until the turn.

The other hands that have us beat are 99 and KK, which is 6 combos total. But they need to be discounted because he often will have played them faster (especially KK before the flop). So call it 3 combos, although I suspect that is an overestimate. (Then again, this isn't a bad flop for slowplaying.)

That makes 4 hand combinations that beat us total.

Combos we beat

The most obvious hand that we still beat is 33. I don't really see how we can assume that the opponent would slowplay 99 but *not* 33. (If you're going to discriminate about which sets you'd slowplay, the logical one would be KK since your opponent is less likely to have a made hand in that case). Perhaps there are some players out there who would slowplay 99 but not 33, and perhaps *you* would play them that way, but we don't have anywhere near enough information to be that particular about *this* opponent's play. So discount the 33 combos by 50%, as we did 99 and KK, which works out to 1.5 discounted combinations.

Accounting for these hands only, we're good 27% of the time, which would make the call wrong (but not terrible).

However, what really swings the math is some of the other combinations the opponent *might* play this way:

55- Nobody mentioned 55 as a possibility, since you wouldn't expect an opponent to make it to the river with that hand as a default. But the opponent may be a calling station. Or he may decide he wants to continue on with a hand with some showdown value depending on how our 2-barrel/3-barrel dynamics have been working -- our hand is actually not all that well defined here. If 55 has made it to the river, it will certainly put in a big raise once it makes a set. So the 3 combos must be discounted fairly heavily, but they're probably worth 1/2 a combo or 1 combo, I think.

AA - If we're allowing for the possibility that our opponent slowplays things in general, including potentially KK before the flop, then AA needs to be considered too. More broadly, both good and bad players play AA in all sorts of weird ways, and when they play it weirdly they often overplay it. AA is a plausible hypothesis surprisingly often when the opponent's play is otherwise strange but he wants to get a lot of money in. There are 6 AA combos -- we certainly need to discount them quite heavily but call them 1 combination after doing so.

AK - While playing AK in this way does not seem very optimal with these stack sizes, it is one of the liver hands with 12 combos. Even if you discount them by 95%, that works out to 0.6 combos. Because there are so few *winning* combinations he might have, that actually makes a material amount of difference in calculating our equity.

Two Pair - This is not a very good board for two pair given his preflop range. Nor is it clear that if he had two pair, he'd play them in this way. Still, an opponent who plays Kxs before the flop -- and some opponents will even if "we" don't like that hand -- will have made two-pair with K9s, K8s, K5s and K3s. 98 is less likely because the 8's are extremely dead. Still, the two pair hands combined were worth something like 1 combo when I summed the estimates of the probabilities up.

As I said, I think it's these "exotic" hands like AA/AK/55/two-pair that really make the decision pretty clear in my view. The possibility that he'd play any *one* of these hands (e.g. exactly A A ) in this way is fairy low. But it's certainly not zero, given that we have a fairly uncertain read on him. I've given you fair amount of information about the opponent and we've formed some suppositions about his play, but I think we're being a bit overconfident about our read on him if we're entirely eliminating hand combinations like AA or 55.

Finally, there is the possibility of a bluff. This is more of a garbage-in, garbage-out assumption since we don't really know that much about his long-term bluffing frequency. But there are a fair number of hand combinations he could get to the river with, and if he bluffs with them even 1% or 2% or 5% of the time, they start to weigh into the equation a little bit given how few winning hand combos he has. I don't know that our hand would be worth a call if it were solely a bluff-catcher, but it certainly gives us a bit of extra wiggle room if we think the decision is otherwise close.

The moral of the story: I suspect *most* poker players are quite good at estimating these hand ranges *most* of the time. But I'm sure we also have some blind spots where our heuristics fail us, and that we tend to err on the side of narrowing the opponent's hand range too much. I know I started to have a lot of trouble in the mid-high limit games when the good players started opening up their games a lot, and the instinctual hand ranges I had developed after a couple of years of playing in the games were thrown off.

This is in accord with people's general tendency to be overconfident when they make predictions, which is a big theme of the book. It seems a lot of what people like durrrr and Galfond did was to detect and exploit cases where people's intuitive estimates of hand ranges are off-kilter.

The answer to the question is in how often he does something crazy on the river. I would assume a jam from this player is a weaker hand/air waay more often than AK or AA or whatever. How often does he jam JT/9T/TT/QT/Ax/44 etc., nobody knows. If he starts with 100 combos of these it does not mean he shows up with 4x more buffs than if he starts with 25 combos.

What is known is that for a few hours at the table no reason has been given to assume he is doing stuff like this, that is why all the live pros who are forced to grind these games say thinking you are good 50% is insanity.

I was discounting half the oversets, (6 combos/2=3 please correct me if I'm wrong) more than half the undersets (i.e. 33, let's ignore 55, so 1 combo of 33, so 4 combos so far) and the one combo of 67cc. That's 5 combos total, 4 of which we lose to.

You have played poker with this individual for a few hours of your life. I agree he's very rarely bluffing. I agree he's very rarely value shoving worse, if ever. Point is he only needs to do so a miniscule percentage of the time to make this a call.

You would need a very very very strong, and scary accurate read to make this a fold.

If you have this read after playing poker for like 100 hands with another person you are reading his ****ing mind. That's my point.

I would not be shocked to lose, I would expect to often. Folding here would be insane.

I agree with most of your general thought process but I think your frequencies are way off.

I am not saying he would slowplay 99 but not 33. I am saying he would slow play both of them an undefined and roughly equal amount of the time, and he will raise 99 on the river much more than 33. 1 combo of 99 and a half combo of 33 (if that) is pretty fair imo.

This is just the way people play... also, fundamentally it's sometimes correct.

99 beats 88 (and 55 and 33), 33 loses to 88 (and 99, and 55, and KK, and 67cc, and the old nit starts to get worried and takes the cheaper showdown,) relative hand strength of these hands is different, and it matters more and more the narrower these ranges get.

Nobody mentioned it's a possibility because it's rare. I also wouldn't discout this entirely, I'd say you can include 0.01 combos of 55 "to be safe."

Again, I agree with your general thought process, and I think your on the right track, I'd just argue that 95% is not even close to accurate. Over the last 5-6 years, I've played countless hands against old guys with AK who had TPTK, I've seen a move like what you are suggesting "maybe" once, probably never. Think over all the hands you have played, or watched, have you ever seen anything like this? 5% (.6 combos) is WAY too generous. If I had to quantify AK here I would say 0.05 combos.


Your correct in saying he might have two pair going into the river, but the problem is he will be extremely worried about raising this hand (for similar reasons as raising 33, obv) I'm not sure where you got 1 combo from, I'd say he thugJAMs 2 pair on the river less than 1% of the time he has two pair on the river facing your line. I'm not sure what this comes out to in terms of combinations but 1 combo is way too much. 0.15 is more likely?

Bluffs - 0.25 combos "to be safe"

This is exactly my point! Everybodies intuitive ranges are way off kilter here, we are just lazy and throwing in all these combinations because that Rusiian guy jammed top pair here vs me last week, and there was a drunk guy who played AK like this 8 months ago.

I would argue that you guys are being overconfident in your assumptions about all of the fluff in his range, not me. Or I would argue that we are equal opportunity "overconfident." In other words, we are making educated guesses, and thats how you solve problems in this game.

Andrew,

The thing is, we're not defining the player as a nut-peddler. He's an older guy. But he has some money and is very comfortable playing in this game, even if he's not the best player. We think he plays a little passively postflop and maybe also a little loosely ... but those those aren't particularly strong reads. If you re-define the problem so that he's someone who's scared to back a set of 3's with his stack on this relatively dry river board ... then sure, it's a fold, I guess. But that's not the type of player we're envisioning here.

As for the fact that it's unusual to see someone playing AK or 55 this way ... you're right that it's rare. The the number of hands you'll observe in a year of playing poker where (i) you raise (ii) there's someone with AK behind you; (iii) there's a board texture something like this; (iv) he decides to play AK this way; (iv) the hand goes to showdown; (v) he shows you his hand ... is going to be very small. But *every* possibility in his hand range is rare for some reason or another. The one possible combo that is playing textbook ABC poker with these actions on each street is 7 6 , but that's just one possible combo.

Still, I see where you're coming from and none of this can be conclusively proven or disproven. My *guess*, though, is that if you did some data-mining and looked at how relatively unknown opponents (say those with <200 hands in a PokerTracker database) played their hands in different postflop spots, unusual lines of play would come up more often than you might expect. A LOT of the profit in poker (and I have done some data-mining on this for the book) comes from the "long tail" of players who play just a few times a year and often play quite oddly or badly. Or we can assume this opponent's play is more sophisticated and optimal, which reduces some frequencies but increases others, especially bluffing.