NLHE Theory + Practice: Week 5

Firstly I'd like to apologize to those of you waiting pateintly for the next sectrion to start. I've ben fairly busy and haven't got round to doing this until now, so sorry for the delay.

This week we'll look at Pages 75-90. This covers the famous "betting for information" idea and then moves on to talk about betting for value on the river. Muchof the advice given here imo is pretty bad, vague and unapplicable to a lot of micro online games. I'll leave you guys to disect what i mean and ciritically assess some of the stuff said in this section.

If we find the book falls short of being useful here, we'll again call upon hands where we have a river sizing problem for value, or when we have the nuts.

To start things off on a flame Sklansky note, I think his example with JT on JJT is pretty retarded, and I fail to see any cirumstances in common situations where checking flop and turn on a board where pocket pairs and also draws will be putting in money. The author seems to advocate failing to build a pot which is one of the worst mistakes you can make in your games with the nuts. This isn't a board where we totally cropple the deck since there are comboinations like KQ, Tx, 88 99 98s etc that may be more inclined to put money in on early streets but not when they brick the river - where the author is recomending a value bet.

What do you guys think about his notion that in spots where we reach the river in a small pot with the nuts, it is better to make a massive raise since our opponents range is limited to bluffs, thin value bets and slowplayed monsters. The first 2 here will be flding to any raise size, argues the author, while the monsters will be folding to none. Therefore it can be better to shovel to target the only hands we can get any value from.

Anyone ever had a spot like this? Examples from your sessions welcome.

This week I really hope we can get some example hands flowing to analyse and examine these concepts through. I have a good one to post for river bet sizing that I will post with detailed analysis when I have more time. I'd encourage y'all to look out and bookmark HHs where we have a river sizing spot, or where we have the nuts on the river. If anyone finds a spot where it is actually profitable to bet just for informatiomn and not for overriding reasons then I'd like to see it!

Gogogogogogogo!

i too have been slack and i'm already one chapter behind. i'll make it my personal drunken mission to be up to date with this before #6. no promises though, i'm only (a lazy excuse for a) human.

thanks for getting this rolling again carrotz...while i agree with your point about building a pot with the nuts, i do not think that was the point of showing the hand, it was more like a "this is what happened, what do we do now " thing, not a "we played this hand super standard well, now we are at the river whats next" but either way i see where you are coming from. I think that he really does a good job of illustrating how nuances in the other guys range (if hes bluff raising or raising with a better hand) can dramatically affect what is the optimal course of action. The section serves as a great advocate against going into auto pilot or robot mode and staying alert and attentive at the table.

I am about to have a sesh so i will post something later with an actual hand

I think this is one of those sections in the book where you need to read it about 2-3 times and filter out alot of shit to really get something out of it - but it can be done.

The whole section on information betting seems useless at face value.

He has an example on pg 79 about it being cheaper to bet 30 witha mediocre hand and give up when called than it is to bet 15 and get called then to bet 60 on the turn into an unknown range. This seems silly though, since we may actually take it down more often with that 60 bet in the first place making a double barrel using a small initial bet being more profitable than a larger single barrel anyway.

In addition, these lines in general seem to be identical to lines we'd take when we were bluffing...ie betting larger to get more folds or double barreling to fold out weak parts of a range. But the whole point of betting a medium strength hand anyway is to get value from weaker hands and not fold them out. So its pretty confusing that our goal with that type of hand is to generate folds rather than calls just because we get a better picture of villains range.

But then again, making small bets with small hands seems folly anyway since that gives out information that our hand is weak anyway. So i dont necessarily disagree that betting larger is wrong...just that his reasoning seems off.

One thing i did get out of that section though is that we should almost never play in a way that allows our opponent to narrow down our range precisely. He talks about this somewhere else in the book where its more dangerous when we have a narrow range since our opponents play better against it...but basically if we play aces in such a way that kings can fold then we clearly messed up somewhere. I guess this leads into the idea of polarized and merged ranges to counter giving up information and allowing opponents to play perfectly but i think that should be for another section...idk.



As to nuts on the river: im not convinced. its been shown earlier in the book that when our opponent calls a super large bet a small amount of the time then that bet might be better than a much smaller one that gets called much more often. However we dont actually know how often hes gonna be calling a shove. He lists hands like TT, J7, 77, and 33 that could be here and would call a shove...but that amounts to 10 combinations, and hes gonna be folding 98 and Jx a bunch where as they might have called a large bet that wasnt a shove.

Basically his reasoning seems to be that in a multiway pot when flopping the absolute nuts, its better to just check it down to allow someone else to hit a 2nd nut type hands with some unlikely combination. However as carroters said, 98, kq, jx, tx and other hands will gladly put in a bet or two...and with a 100bb stack as typical of an online game, itll make the spr go down pretty quickly. Surely the value we get from shoving the river in a low spr pot and getting 98 and jx to always call in addition to the extra bets from kq and such exceeds that we get form someone making an unlikely 2nd nut hand.

Also it would seem that we'd only take the c c shove line with a nut type hand anyway so we'd give information about our hand and allow our opponent to make hard laydowns easier...seemingly going against the advice in the previous section which seemed to be not giving info about our hand.

His general point of raising super large when our opponent is never calling a raise except with a monster seems valid though...just not in the case were we're multiway on a somewhat wet board.

My computer at home died on me so I can't post a hand until I get a new powersupply unit, which could be a few weeks. So for this weeks discussion I will simply post my thoughts and ask questions when others post hands.

Yeah I'm gonna let that JsTs on JJT board example slide because he specifically states it would only happen under certain circumstances (I'm not quite sure what those circumstances would be at the moment, but the flop and turn play don't make a difference with what he is talking about and it could have just as easily not been mentioned at all). And with everything else I disagree with about this particular example, he states that its not always going to be this case.


To be perfectly frank I dont understand his examples of quantifying the information at all. On page 76 all I see is him calculating the EV for calling an all-in. Great, he read spoonys "calling all-in" thread too. You don't gain any information if you already know villains range (99+,AK,and AQ). All you gain is the knowledge of what they had THAT particular time. In the other examples it seems that paying for information gives such spacific information that it would only be profitable if you play that villian a bunch.

As for value betting the river, mathematically it makes sense to me but seems counterintuitive to experience (severely limited as it may be). Ill play around with the math once I can review old sessions again.

Originally Posted by JKDS

His general point of raising super large when our opponent is never calling a raise except with a monster seems valid though...just not in the case were we're multiway on a somewhat wet board.

this.

like i said before i do not think that was a guide on how to play a monster hand, more a hand that was the absolute nuts that had been checked down and what should be done on the river in that situation. But i agree there are def pieces of the board that will give us value like you said 98 and shit like that.

anyway as i said i would do before, here is a hand where i was faced with a river situation similar to what he was describing (i think).

this hand is against a complete unknown, he had not played in the 3 hands before this, but looking at his stack size and his minbet preflop, i assume he is pretty fishy


Full Tilt No-Limit Hold'em, $0.05 BB (9 handed) - Full-Tilt Converter Tool from FlopTurnRiver.com

Hero (CO) ($4.93)
Button ($5.24)
SB ($1.77)
BB ($5.20)
UTG ($2.53)
UTG+1 ($5)
MP1 ($5.12)
MP2 ($5.79)
MP3 ($3.86)

Preflop: Hero is CO with A, K
4 folds, MP3 bets $0.10, Hero raises to $0.25, 3 folds, MP3 calls $0.15

when he calls i have his range as Axs,ATo+, 34s+, j9s+,kts, kj+,22+, although i think he would 4b aa kk qq aks.

Flop: ($0.57) 5, J, A (2 players)
MP3 bets $0.57, Hero calls $0.57

not sure how to interpret the donk, against any thinking player i would assume it is for value but with someone who is a potential fish who knows? i think his donking range is probably diamond flush draws, Ax, TT, AJ, JJ, JT,JQ, 55, maybe J9s 88, 99, KQ, KT, KJ. So obviously i am ahead of that range so i call for another street.

Turn: ($1.71) J (2 players)
MP3 checks, Hero checks

when the Jack falls i feel like the turn check is a trap. In hindsight i should have bet to price out all the draws out there but i felt like checking behind to keep the pot small.

River: ($1.71) 2 (2 players)
MP3 checks, Hero bets $0.80, MP3 raises to $3.04 (All-In)

Total pot: $7.79 | Rake: $0.51


I actually am having a lot of trouble analyzing this river. When he checks i should probably have checked behind, i get value from Ax, and maybe 99 or TT, but thats it, all his missed draws fold for sure, and he maybe bluff raises. When he checks the river i take all the jacks out of his range.

As you can see, i bet .80, a little under half pot, and he raised me all in. Here i figure his range is pretty polarized by monsters and air. Monsters less likely because given the course of the hand i do not seem too likely to call much of a river bet at all and he checked it twice after the second jack hit. Seems more like missed flush draws to me so i called, he shows AThh and i win, but still it was a tough call, and id like to hear some more thoughts on the river if anyone would be so kind.

I sort of get what says with the JT on JTT - our opponents range is heavily polarized between monsters that he's been slow playing and weak pairs/bluffs, very few good decent made hands so we just want to focus on the nut part of his range however I don't think shoving is best here since we wouldn't shove the river with air but we might bluff raise smaller.

philly in your hand I'd 3 bet bigger since he's likely to call with many dominated hands, need to think about the rest of the hand before I comment.

Philly - Defo iso bigger preflop and make him make more a mistake by flatting your 3 bet too wide with whatever shit he wants to do this with oop.

I don't mind flatting the flop here, but I think raising with the ointention of being able to get it in over the turn and river on safe cars is maybe better. Thing is I doubt many fish are just donking pot with a better hand here like ever. So his range is likely to be made up of worse aces and draws that you want to get to put more money in with now. He can have air but with no reads that he's super bluffy I'm happy to just raise flop and get it in vs this guy.

On the turn you need to bet here 100%, not doing so is a fairly big mistake. His mosy likely hands are still draws and inferior top pairs that you can win a sizeable pot from by betting turn and betting river. Checking back just gives him a free cardfor no reason and loses a street of value from Ax.

Bet the river bigger for value, again you're just leaving money on the table since Ax is calling like any size up to pot, and he doesn't have a ton of weak hands that are calling this but folding to anything bigger. His river shove doesn't seem to make a whole lot of sense, but I suppose he can play a Jx this way. Given all the draws brick and his weird line here though, I think we can stick enough bluffs in for this to be a call.

Looking for hands where we are the ones looking to bet the river for value with a monster, so we can disect villains calling range to various sizes and decide what's best.

OK, here's an example from my session today. The villain is basically an unknown, I've only been at the table for 6 hands and he's running 33/17.

On the flop, calling seems pretty straightforward -- I have the NFD and two overs, and I don't really have him on a range since he's unknown. As a guess, he could have air and be trying to steal this pot, a pocket pair, a set, two pair, paired the board, a worse flush draw, or a straight draw.

In retrospect, I wish I raised the turn instead of just flatting. I have the best hand and should have tried to get it in while he was still betting. I now weight the hands towards the draws and sets, and maybe hands like TT-QQ with a club, because I can't imagine that he would be doing this with any made hand less than a set on this kind of board. Honestly, whatever he has, this is a weird line to me.

I'm not sure about my sizing on the river bet -- comments?


Full Tilt No-Limit Hold'em, $0.05 BB (6 handed) - Full-Tilt Converter Tool from FlopTurnRiver.com

BB ($4.82)
UTG ($5)
MP ($5.07)
Hero (CO) ($5)
Button ($5.41)
SB ($5.94)

Preflop: Hero is CO with ,
2 folds, Hero bets $0.20, 2 folds, BB calls $0.15

Flop: ($0.42) , , (2 players)
BB bets $0.42, Hero calls $0.42

Turn: ($1.26) (2 players)
BB bets $1.26, Hero calls $1.26

River: ($3.78) (2 players)
BB checks, Hero bets $2.20, 1 fold

Total pot: $3.78 | Rake: $0.25

i raise that flop and that turn...river bet i would bet more also. youll get a lot of value from sets and tt-qq, maybe a9.

ok so here is an example of the analysis i think that they are looking for in this chapter.

My villain is 40/17 over 35 hands, i have notes that he likes to chase gutshots and limp/calls wide


Full Tilt No-Limit Hold'em, $0.05 BB (9 handed) - Full-Tilt Converter Tool from FlopTurnRiver.com

UTG+1 ($5.24)
MP1 ($10.08)
MP2 ($8.47)
MP3 ($11.07)
CO ($6.32)
Button ($13.37)
Hero (SB) ($4.95)
BB ($3.34)
UTG ($4.86)

Preflop: Hero is SB with A, J
3 folds, MP2 calls $0.05, 3 folds, Hero bets $0.20, 1 fold, MP2 calls $0.15

Flop: ($0.45) 8, 3, 2 (2 players)
Hero bets $0.20, MP2 calls $0.20

Turn: ($0.85) 7 (2 players)
Hero bets $0.60, MP2 calls $0.60

River: ($2.05) 5 (2 players)
Hero bets $2.05, MP2 calls $2.05

Total pot: $6.15 | Rake: $0.41

ok my line is whack and i should have bet more on the flop cause im still getting value from trips, 8x, gutshots and kx of clubs. That being said my weak bets enabled me to get to a river and set up a perfect situation to analyze bet sizing.

His range when we get to the river is Kc8+ (non flush), 86+, 99, TT, 69s, 46s, a4, 87s, 88, 77, 55, 33, 22 smaller flushes like QTcc, QJcc, JTcc, T9cc, 45cc, 56cc, 67cc, 79cc, 7Tcc. j9cc, and some busted draws like jt 9t j9 with 1 club

for a total of 18 Kc combos, 87 8x type hands, 12 99-t hands, 20 straights, 2 two pairs, 15 trips, and 10 flushes, and 21 busted draws

in all probability he is not playing flushes this way so we will cut that amount in half, so 5 flushes, bringing the grand total of combos to 185

lets say i have 4 options.

1. Check
2. 1/2 pot bet
3. pot size bet
4. Shove (basicall 2x pot size bet)

1. Ok for the sake of simplicity we will say when i check to him he checks behind 75% of the time, and 25% of the time he throws out a pot size bet. So when i check, 75% of the time i win 2.05, and 25% of the time i win 4.10, averaging out to- 2.56

2. We will say he calls a half pot bet with his entire calling range, and fold out his busted draws like KcX J9 or JT or T9 with a club. But will call with anything else in his range. So we make 3.08 when we win and 2.05 when he folds. The part of his range he is folding is 31/185, or 21%. So 79% of the time we win 3.08 and 21% of the time we win 2.05, for an average of 2.86

3. He is calling a pot size bet with his whole range from section 2, except for like half of his 8x range, limiting it to 8j+ which is 42 combos. so he is folding 76/185 which is 41 % of the time. So 59% of the time i win 4.10, and 41% of the time i win 2.05, averaging out to be- 3.26

4. And when i shove we will say he is only calling with 88,77, straights and flushes. which is 31 combos. 31/185= 16.7 of the time he is calling my shove. So i make 6.15 17% percent of the time and 2.05 83% of the time, averaging out to 2.75

as i have shown in my analysis the pot size bet was the best option.

In order to make a shove the best option you just set up this simple equation

6.15 (pot when i win shove) * X (percent of time he calls as a decimal) + 2.05 * (1-x) > 3.26 (the amount i made off the pot size bet)

so 6.15x+ 2.05- 2.05x > 3.26

4.1x > 1.21
x> .295

so i need him to be calling my shove with 29.5% or more of his range for this to be more ev+ than the pot size bet.

Hate to be discontinuous, but, in regards to JKDS' earlier post:

Originally Posted by JKDS

(...)
Basically his reasoning seems to be that in a multiway pot when flopping the absolute nuts, its better to just check it down to allow someone else to hit a 2nd nut type hands with some unlikely combination. However as carroters said, 98, kq, jx, tx and other hands will gladly put in a bet or two...and with a 100bb stack as typical of an online game, itll make the spr go down pretty quickly. Surely the value we get from shoving the river in a low spr pot and getting 98 and jx to always call in addition to the extra bets from kq and such exceeds that we get form someone making an unlikely 2nd nut hand.

Also it would seem that we'd only take the c c shove line with a nut type hand anyway so we'd give information about our hand and allow our opponent to make hard laydowns easier...seemingly going against the advice in the previous section which seemed to be not giving info about our hand.

Completely agree on the first point about how we are better off extracting value from hands like 98 and Jx, rather than waiting until the river to unleash a huge bet on the slim hope that one of our opponents has improved to the 2nd nuts (or some other very strong hand).

One thing that really struck me when I read this section was that this entire line described by Sklansky - the whole c, c, shove deal - is something that seems to happen with surprising frequency in the microstakes. I'm sure we can all remember situations in 2NL where a pot was checked down to the river, the last betting action having taken place preflop, only to witness the villain either dole out a large bet on the river or even go all-in. Granted, I would posit that a huge portion of the time, these players are not holding the nuts -- and at least some of the times they do have very good hands, that they would have actually just improved to them on the river -- but I get the impression that these types of lines just come off as being weak attempts by the Op to try and recapture value lost on previous streets. Something that any astute player would have avoided in the first place.

Bearing all this in mind, there is probably some validity to the theory in the sense that, ignoring previous action, if we are holding the nuts on the river and our villain's range tends towards either weak made hands and a few slow-played monsters, we may be better off betting large or raising for value in order to maximize our expectation. This was not made very clear within the text, however.

$0.02/$0.05 No Limit Holdem
8 Players
Hand Conversion Powered by weaktight.com


Stacks:
UTG ($3.02)
UTG+1 ($2)
Hero ($16.77)
MP2 ($12.95)
CO ($3.07)
BTN ($2.93)
SB ($12.08)
BB ($4.43)

Pre-Flop: ($0.07, 8 players) Hero is MP1
2 folds, Hero raises to $0.20 ($16.57), 2 folds, BTN calls $0.20 ($2.73), 1 fold, BB calls $0.15 ($4.23)

btn 34/17, bb is a 16/7 who loves calling my raises with Broadway type hands and likes leading with good hands (like TP on a dry board)

Flop: ($0.62, 3 players)
BB checks ($4.23), Hero bets $0.30 ($16.27), BTN folds ($2.73), BB calls $0.30 ($3.93)

nuts! ok pot building is absolutely necessary...question is how.

villains have 55bbs and 85bbs before flop action. hero bets half pot

thoughts:
1) if the pot stays multiway, they arent so deep that ill have trouble lowering the spr
2) bb is unlikely to have a set imo, and sets will raise a fair amount of the time cuz they hate seeing spades fall. They have 7 outs to improve on the turn so i should stay ahead alot even if they do end up calling with sets
3) Tx, the Js and the Qs likely fold to larger bets. Ks never folds.

So basically im thinking that i can bet small in order to keep worse hands in with little risk, meanwhile those worse hands have a small chance of improving to beat me


Turn: ($1.22, 2 players)
BB checks ($3.93), Hero bets $0.75 ($15.52), BB calls $0.75 ($3.18)

bad card, but i really doubt the bb plays a set this way. his range is mainly flush draws, tx, and like JJ/QQ sometimes. maybe bet larger...but i really dont want him to fold these marginal hands that have little chance to improve. i guess Tx has like 10 outs :/

River: ($2.72, 2 players)
BB checks ($3.18), Hero bets $3.20 ($12.32) ...

really doubt he has a nut hand now...and i feel the only hands left are either calling any bet or folding...so overbet.

thoughts?

JKDS, I'd bet the flop a bit harder probably to .45 or .50. reason being, Ireally don't think either will fold a T or better for this sizing very often at all and probably not Qs+ either. There are also a bunch of hands that think they have decent equity yet obv don't at all like 8s8x 9s9x etc. So yeah there's plenty of hands in their range that you can afford to build the pot a lot more vs before the 4th spade comes to slow down your action form Tx etc.

On the turn, again, your bet size is too small. Tx is certainly never folding no matter how close to pot you make this and marginal pairs are likely folding to any size if they don't have a spade.

On the river I agree with your analysis, 8s8x and that that junk is now not calling all that often and there's not all that many combos of these hands anyways. Tx is a sizable part of his range and defo not folding, since he flats so many boradways I think Tx is enough of his range for you to mainly be targeting it for value and I like the slight overbet. If you build the pot a bit more though on previous streets you can really increase your EV for the hand though.

Ya I see your point. If by just betting a lil more on the flop i can make my turn a bit bigger and then the river could be a less than pot shove, as well as making sets play in a more defined way (probably raise more often if i bet bigger) i think its worth it. Hes a bit stubborn with me and acts sherriffy so betting more is definitely best. kk

Its actually pretty damn tough to find a hand where overbetting the river with the nuts good lol

Something I think are vague and pretty annoying in the final thoughts part of this chapter.

"If you are last to act and might be beaten, your bets should tend to be smaller than they would be if you were fairly sure you had the best hand.

Okay this is pretty misleading. The author is suggesting that how likely it is that you are beaten should determine how large you should bet on the river. Umm no...the factor determining how much you bet on the river with a marginal hand for value is not how many better hands are in villains range, but how much equity you have vs his non folding range/continuing range for various bet sizes and how these different potential bet sizes change his continuing range and hence the EV of the bet. If you are "likely to be beaten" you may well not even have more than the 50% equity needed to bet vs his continuing range and shouldn't bet at all since he'll be folding a lot of weak hands usually.

EG. You are against a passvie villain who cannot fold much at all on the river with a decent piece of the board or pair. You have AT on QT224 and have bet the flop and turn for value. On the river you decide that hes calling with anything 33+ but is passive enough to have all Q combos that he reaches the flop with. He cannot fold these 1 pair hands to a psb. So lets say at an estimate that we have 63% equity vs villains continuing range, as there are more combos of 33-99 and Tx than Qx. So should we bet less on this river than we should if he had 0 Qx combos in his range? Of course not, in both cases, he calls just as often to the psb as he does when we bet less, and so, if betting at all is +EV, then betting the pot will be most +EV in either case, regardless of how much of the less than 50% part of his range is beating us. (assuming here that a psb is his inflection point where he starts to fold a lot more often as we bet more than this and hecne our EV goes down.)

So yeah, vague misleading statement - EV vs a continuing range is what decides how much we should bet on the river, not the amount of stronger hands in villains range as a whole.

Challenge: Post a hand where you bet the river for value and have a go at estimating which size is better out of 1/2 pot, full pot, and 2x pot. Make a continuing range for villain for each bet size and work out which range you have the most equity against using pokerstove.

If you can't find a hand like this, make one up. It's great to practice assigning continuing ranges vs your value bets. Get in to the habit of doing this in the right way and not going "meh I'm beat here quite a lot so I'll bet a bit less."

One thing is true, the weaker the range you are trying to extract value from, the less you should bet in general (not always) The more hands that beat you, the weaker the range you're trying to extract value from. This is likely what he means to say, but yeah felt it needed a lot of clarification for beginers so you can practice doing this in the right way.

/rant

gogogogo.

If feeling generous, you could interpret the author's statement as a badly worded and unclear version of that.

i already did that?

Finally got around to this, here's the hand.

weaktight | Hand Poll | AA - $0.05/$0.10 No Limit Holdem

Opponent is an unknown, I know that it's impossible to bet 2x pot here but I'll do the calculation for a shove.

1/2 pot continuing range {J8o+, 88-TT, A4o} 79% equity.

Pot {J8o, JQo+, 88, A4o} 62.5% equity.

Shove {J8o, JKo+, 88, A4o} 50% equity.

I included A4o to account for the possibility that he may slow play a turned straight but sets probably raise before the river. So I think that 3/5 pot is a good compromise size since most Jx will still call.

Comments appreciated.

Originally Posted by HoopyDude

Finally got around to this, here's the hand.

weaktight | Hand Poll | AA - $0.05/$0.10 No Limit Holdem

Opponent is an unknown, I know that it's impossible to bet 2x pot here but I'll do the calculation for a shove.

1/2 pot continuing range {J8o+, 88-TT, A4o} 79% equity.

Pot {J8o, JQo+, 88, A4o} 62.5% equity.

Shove {J8o, JKo+, 88, A4o} 50% equity.

I included A4o to account for the possibility that he may slow play a turned straight but sets probably raise before the river. So I think that 3/5 pot is a good compromise size since most Jx will still call.

Comments appreciated.

you arent looking for the spot you have the most equity against his range, you are looking for the most expected value,

so you have to figure out how much you win the 80% of the time he calls, and how much you lose the 20% just to find out the ev of the 1/2 psb. You have to do the same for all 3 in order to figure out the most ev + play

^^^ Yeah this is what I meant to say, sorry should have beeb a bit clearer. You will almost always have less equity vs a calling range when you increase the size of the bet, or certainly no more equity.

Finding out the EV of eachbet is what allows you to decide which is best.

Ah I get what you mean.
1/2 pot EV = 0.79(7.50) - 0.19(2.50) = $5.45
Pot EV = 0.625(10) - 0.375(5) = $4.38
Shove = 0.5(13) - 0.5(7.75) = $2.63