Buzz - Backdoor low odds for Omaha 8b!

[Big Dave D]

Hi Buzz

I remember u putting some analysis together, either here or on rgp about backdoor lows....could you please repost it?

cheers

Dave D

[DPCondit]

I'm not sure what Buzz has posted on the subject, but it is a simple calculation.

If you have A [img]/forums/images/icons/spade.gif[/img] 2 [img]/forums/images/icons/club.gif[/img] K [img]/forums/images/icons/diamond.gif[/img] T [img]/forums/images/icons/heart.gif[/img] for example, and the flop is 5 [img]/forums/images/icons/spade.gif[/img] J [img]/forums/images/icons/spade.gif[/img] J [img]/forums/images/icons/diamond.gif[/img] , you have seen 7 cards, 52 - 7 =45 unseen cards, out of the unseen cards, the ones that help your low are 3, 4, 6, 7, and 8, four of each, so 5 x 4 = 20 cards that help your low on the turn. 20/45, Since you need to hit the turn and river, any low card that hits reduces your helpful low cards by 4, 20 - 4 = 16, so 16/44 x 20/45 = .16161616, or about a 16% chance of making low, or 5.2 to 1 against.

Now it's a little trickier.

So say 5 opponents, 12 bets in the pot, first player bets, two people fold, and two people call to you.

If you figure 3 callers (including you, x 1 big bet each on the turn and river each), lets figure it out.

12 bets + 4 bets, plus 6 big bets (12 small bets), so 28 small bets on the river.

Out of 1000 times, you will fold 556 times on turn and lose one small bet (-556).

Out of 444 times, you will fold 283 times on the river and lose three small bets 283 x 3 (-849).

161 times you make your low, there is 28 small bets/2 = 14 small bets. If you figure you have about 90% of the low equity at this point (adjusted for possible splits), then .9 x 14 = 12.6 minus the 5 bets you put in from the flop forward, so 12.6 - 5 = 7.6 small bets, so 161 x 7.6 = 1223.6.

So over a thousand tries we lose 181 bets, so even with 6 callers, and a raise preflop, we still lose money here.(1224 - 849 - 556 = - 181).

Buzz may have something to add to this.

Don

[Buzz]

Hi Dave - I see that Don Condit has responded to your request. I'm going to post what I have already written, and then I'm going to study his response.

I don’t have a file labeled backdoor lows. I’m not sure what I wrote about them in the post to which you are referring. However, it’s not hard to put something together for you. There would be three simple cases: (1) you have two low cards, (2) you have three low cards, and (3) you have four low cards.

(Case 1) After the flop you have two low cards. For example, you saw the flop with A2JJ-non suited and the flop is 39K (suited or not). In this case, you have a poor shot at high but you have the nut low back-door draw but no counterfeit protection.

To make the nut low, you need one of the following two rank combinations: 45, 46, 47, 48, 56, 57, 58, 67, 68, or 78 on the turn and river. Since we’re dealing with combinations, the order of the cards is unimportant. In other words, 4h5h and 5h4h are the same combination. It doesn’t matter if the four or the five comes first.

Note that there are 10 two rank combinations and no other combinations that will give you a low. Each of these 10 two rank combinations can be made in 16 ways. Here are the 16 ways to make 45: 4s5s, 4s5h, 4s5d, 4s5c, 4h5s, 4h5h, 4h5d, 4h5c, 4d5s, 4d5h, 4d5d, 4d5c, 4c5s, 4c5h, 4c5d, and 4c5c. Thus there are 16*10 = 160 two-card combinations (out of 990 possible) or ways to make low. The odds against your making low in the next two cards are thus 830 to 160 or about 5.2 to 1 against.

But in addition, if you have A2JJ and the flop is 29K-rainbow, if you miss a 4, 5, 6, 7, or 8 on the turn, you’ll continue with a jack on the turn.

After a jack on the turn, you would hold A2JJ and the board would be 29KJ. An opponent who started with KQJT, QJT9, QJT8 or A2TQ would have a straight. In a very tight game opponents might not be playing KQJT, QJT9, or QJT8, but you can bet they would be in a loose game. And you can bet that anyone with KQJT, QJT9, QJT8 or A2TQ would continue playing after a flop of 29K. After a jack on the turn, someone with QTXX would already have a straight. In addition, the river could make an additional straight or a backdoor flush possible. Thus catching a set on the turn is fine, but in a loose game you would want the set of jacks to improve to a full house by the board pairing on the river. With a hand/board of A2JJ/29KJ, you would have ten outs on the river to make a full house for a probable scooper. Of course if you caught the case jack, you’d be golden, and if the board paired with twos or nines (6 outs), you’d also be highly favored. But if the board paired with kings, about half the time with your underboat (jacks full of kings) would get burned by someone with quad kings or kings full of jacks, nines, or twos.

If you catch a jack on the turn, you’re looking for the board to pair on the river. But the jack has to come first. When the order of the cards matters, we have to deal with permutations instead of combinations. We have 2*10 = 20 permutations, JJ, J2, J9, and JK. These 20 permutations are for the nuts, but as noted above, they don’t always all win. However, for now let’s put the possibility of some of these combinations not winning aside.

Thus if you call a bet on the second betting round, there are a total of 22 cards which will keep you in the hand for the third betting round (after the turn). Let’s deal with the A2 aspect of the hand first.

No one can say how often an opponent who has been dealt A2XX will call a bet after a flop of 39K. However, we can calculate the probability of an opponent also having been dealt A2XX. In a nine handed Los Angeles area game, when you are dealt A2JJ, it turns out one opponent will be dealt A2XX about 38 % of the time and two opponents will be dealt A2XX 5 % of the time. Only 57% of the time will you be the only one who was dealt A2XX in a nine handed game. (This becomes 53% in a ten handed game).

After a flop of 3h9dKc, it is difficult to know for certain the cards an opponent who bets holds. Could be a set of kings or top two pairs. These are both legitimate bets, in my humble opinion, after this flop. Could also be a semi-bluff bet - perhaps KQJT, QJT9, a set of nines, a set of treys, perhaps top and bottom pairs, perhaps even bottom two pairs. Could be perhaps someone with a king, a couple of back-door flush draws, and maybe also a back door low draw. Or maybe an opponent betting after a flop of 39K-rainbow does not even have that much. A bet after 39K-rainbow could be a pure bluff. Kind of depends on the group with which you’re playing. When two of the cards are of the same suit, you also have flush draws who might be betting.

When you hold A2JJ, after a flop of 3h9dKc another player could reasonably bet or call with A2KK, A29K, A299, A233, A23K, A239, Ah2h4dKd, or even A-2-anything-anything. Thus there would be a good chance another player, especially an unsophisticated player, who was dealt A2XX would continue playing after the flop, even though the flop is not favorable for low draws. Thus if the total pot is P dollars, because of the quartering and sixthing effect, you should expect your share of it to be, on the average, 0.57*(1/2)P + 0.38*(1/4)P + 0.05*(1/6)P = about 0.39P. In other words, in a nine handed game, especially with unsophisticated players, because of quartering and sixthing, you’re only going to get about four tenths of the pot, on the average, instead of half of it when you *do* make low. (it’s even a bit worse in a ten handed game).

In order to be getting favorable odds to play the hand/flop, A2JJ/39K, solely on the basis of low, there would have to 15 chips in the pot for every 1 chip you invested on the second betting round. Why 15 chips? I’ll try to explain.

Suppose no one bet after the second betting round, and suppose the pot had 16 chips in it after the second betting round (15 chips + the chip you, yourself, add to the pot on the second betting round).

When you are sole winner for low, you win half the 16 chips in the pot, or 8 chips. When you are quartered, you win one quarter of the 4 chips in the pot or 4 chips. When you are sixthed, you win one sixth of the 12 chips in the pot or 2.667 chips.

On the average, when you do make low, you win 8*0.57 (not quartered or sixthed) + 4*0.38 (you’re quartered) + 2.667*0.05 (you’re sixthed). The total is 6.2 chips. In a sixteen chip pot, that’s your share of the win when you do win.

But one of those chips was yours, the one you put in to call the bet on the second betting round. Therefore what you actually win (when you do make low) from the 16 chip pot is 5.2 chips. Thus going for just the low half of the pot, your pot odds are 5.2 to 1.

Recall that with a hand/flop of A2JJ/39K, the odds against you making low are 5.2 to 1. The pot odds have to be at least 5.2 to 1 (the odds against making low) to justify calling a bet. Therefore, the pot needs 16 chips in it to justify calling a one chip bet on the second betting round.

(If there were 15 chips in the pot and if it cost you 1 chip to see a bet at the end of the second betting round, your pot odds would be 4.8 to 1. If there were 17 chips in the pot and if it cost you 1 chip to see a bet at the end of the second betting round, your pot odds would be 5.6 to 1.)

Thus to get favorable odds to call a bet on the second betting round when your hand is A2JJ and the flop is 39K, solely on the basis of your low draw, the pot has to have 16 chips for every chip you invest on the second betting round.

But there is also the possibility of catching a jack on the turn. In that case, there will be no low and you are playing to scoop with a full house. There are 20/1980 permutations that will allow you to scoop. How much value do they add to the hand and how do we figure it and then combine it with the backyard low draw?

Let’s again assume there is no betting after the second betting round. The odds against making a full house, with a jack coming first, are 1960 to 20 against or 98 to 1 against. To justify a call on the basis of making a full house with the jack coming first, there would have to be 98 chips in the pot after the second betting round for every chip we invested on the second betting round. 1/98 doesn’t seem like much, but it does add a bit to the value of the hand, as follows: 1/16 + 1/98 = 1/13.8.

Thus, taking the value of the pair of jacks into account, and combining it with the value of the low draw, it turns out there have to be a total of 13.8 or 14 chips already in the pot with no possibility of a raise behind you for every 1 chip you would contribute on the second betting round. 14 chips may be a bit optomistic, because of the chance of a full house made with your jacks losing. 15 chips is a bit more conservative. But it's something in that neighborhood. If you figure implied odds, I think you get back to 14 chips.

You could never do that math in your head in a game. You really just have to have an idea of how it all works when you hold A2XX and the flop has only one low card. Unless you have flopped a set or a decent straight or flush draw, the value of your A2 combined with the value of the XX component of your hand will generally require the pot to have something like 14 chips with no chance of a raise for every chip you contribute on the second betting round when it is your turn to act.

14 or more small bets in the pot and only 1 small bet to you at the end of the second betting round is far fetched - but not absolutely impossible. There could have been six players who saw the flop with a pre-flop raise, making a total of twelve small bets in the pot before the flop. Or there could have been four players who saw the flop with a pre-flop raise and a pre-flop re-raise, also making a total of twelve small bets in the pot before the flop. Then after the flop, the player directly to your left (not counting anyone who already folded between you and that person) could bet and one other player could call. You would be looking at a pot composed of 14 small bets and it would cost you one small bet with no possibility of a re-raise. In that case, if your hand was A2JJ and the flop was 39K, you would have odds (barely) to call the bet.

At that point, there will be 15 small gets (7.5 big bets) in the pot. If you miss on the turn, you fold. If you catch 4, 5, 6, 7, or 8 on the turn, you will have 16 outs for the river for low. The board will be 349K, 359K, 369K, 379K, or 389K. thus the odds against your making low on the river are 28 to 16 or 1.75 to 1. Someone with a set of nines or one of the other hands listed above might have bet the flop and someone with a set of kings might have slow played on the second betting round. If the same person bets the turn, the player with the set of kings may suddenly come alive and re-raise here. Otherwise it’s hard to imagine anyone raising on the third betting round, but sometimes there are some real fruitcakes in the game. At any rate, after a 4, 5, 6, 7, or 8 on the turn you have odds to call a double bet (bet and raise) on the third betting round (but not quite odds to call a raise and a re-raise, if you can see it coming from a fruitcake).

If you catch a jack on the turn, you will also have odds to call a bet. You're going to get burned if an opponent has KKXX, but if you're going to play hands like A2JJ and give value to the jacks, you simply have to bite the bullet when you bump into an opponent with a higher pair that clicks with the board when your pair of honors also clicks with the board. It won't happen much.

Gee whiz. Sorry for another longee. I thought this would be easy. but I’m already on page five and haven’t dealt with cases two and three. I’ll cut them short.

Case 2. With A23J, after a flop of 49K, you would need a turn/river combination of A5, A6, A7, A8, 25, 26, 27, 28, 35, 36, 37, 38, 56, 57, 58, 67, 68, or 78 to make low. There are 12 ways to make the first 12 of these and 16 ways to make the last 6 of these. 12*12+6*16=240 two card combinations (out of 990) that would make low for you. Odds against making low are thus 750 to 240 or 3.125 to 1 against. You have roughly the same problem of being quartered or sixthed in a loose game as you did with A2, and again you’re only going for low. Here I think you need at least 9 chips in the pot for every 1 chip you contribute on the second betting round. This is very possible. If you have position, and if there were six players who saw the flop, and if the opening better bets and three intervening players all call, it’s one bet to you with no chance of a reraise, and you’re getting 9 to 1 pot odds.

Case 3. With A234, after a flop of 59K, you would need a turn/river combination of A2, A3, A4, 23, 24, 34, A6, A7, A8, 26, 27, 28, 36, 37, 38, 46, 47, 48, 47, 48, 67, 68, or 78 to make low. There are 9 ways to make the first 6 of these, 12 ways to make the next 14, and 16 ways to make the last 3 of these. 6*9+14*12+3*16=270 two card combinations (out of 990) that would make low for you. Odds against making low are thus 720 to 270 or 2.67 to 1. This time I think you need at least seven chips already in the pot for every chip you venture on the second betting round in order to continue after the flop. That’s almost a gimmee in the ring games in which I play.

In summary, there is hardly any way you’ll have proper odds to draw for low if you are only holding two low cards when there is only one low card on the flop. But if the betting got jammed on the first betting round such that there came to be at least 16 small bets in the pot when it was your turn to act with no possibility of a raise behind you, you’d have odds to play for the low half of the pot. In my humble opinion, it would be crazy to do this without the nut low.

Still in summary, if you have three low cards including the nut low and nut low counterfeit protection, as with A23X after a flop of 49K, 59K, 69K, 79K, or 89K, or as with A24X after a flop of 39K, then if there are 9 small bets already in the pot and you are acting last on the second betting round, then you have odds to call one small bet.

Finishing the summary, if you have four low cards after a flop with only one low card, including the nut low and nut low counterfeit protection, you only need 7 small bets in the pot with no possibility of a raise behind you to call one small bet. I think you’re getting odds to see the turn. Note, as an aside that A23H is a good deal better than A2HH but A234 is not very much better than A23H. A2HH<<<A23H<A234.

I’m sure that’s not the way I answered in the post to which you are referring, Dave, but it’s a reasonable odds-wise approach, I think.

Now I go read Don's response.

Buzz

[Buzz]

“16/44 x 20/45 = .16161616, or about a 16% chance of making low, or 5.2 to 1 against.”

Hi Don - Thanks for pitching in. We did this differently, but ended up with the same 5.2 to 1 against result for making low with a hand having only two low cards when there is only one low card on the flop. Another time and I might have approached the problem the same as you did here.

“ Now it's a little trickier.”

Yes, because you don’t quite know what your opponents will do.

“ If you have A 2 K T for example, and the flop is 5 J J...”

I like your choice of an example better than mine. I got into the sticky problem of dealing with a hand that also had a shot (albeit a poor one) at high. Then I decided to go ahead with it, because that’s what often happens. But your hand/flop isolates the low draw. For your example flop/draw, I figure the pot needs 15 chips for every chip you contribute on the second betting round. Those 15 chips are what everyone, including yourself, contributed on the first betting round + what your opponents (but not yourself) contribute on the second betting round. 15 is the break even point. More than 15 is gravy while less than 15 is a clear fold. Maybe I've stumbled upon an easy approach to figuring when you have odds to call. Not sure if I've made it clear.

“ 12 bets + 4 bets, plus 6 big bets (12 small bets), so 28 small bets on the river.”

Looks like your own bet is included as one of the 4 bets. Thus the total here is 15 bets. Hmm. I have this as the break even point.

"12 bets + 4 bets, plus 6 big bets (12 small bets), so 28 small bets on the river."

O.K.

"Out of 1000 times, you will fold 556 times on turn and lose one small bet (-556)."

25/45 you fold after the turn. -556. O.K.

"Out of 444 times, you will fold 283 times on the river and lose three small bets 283 x 3 (-849)."

(20/45)*(28/44) = .283 you fold on the river. (283)*3 = -848. O.K.

"161 times you make your low, there is 28 small bets/2 = 14 small bets. If you figure you have about 90% of the low equity at this point (adjusted for possible splits), then .9 x 14 = 12.6 minus the 5 bets you put in from the flop forward, so 12.6 - 5 = 7.6 small bets, so 161 x 7.6 = 1223.6.

I'd use 77% instead of 90% here and get ~11-5 = 6 small bets. then
161*6 = +966

"So over a thousand tries we lose 181 bets, so even with 6 callers, and a raise preflop, we still lose money here.(1224 - 849 - 556 = - 181)."

I have -556-848+966 = -438, following your general method, but using ~77% as the low equity. Shoot! Looks like my 15 chip to 1 chip method needs refinement.

Excellent analysis, Don. Thanks again for pitching in.

Buzz

[DPCondit]

Buzz wrote:
"I'd use 77% instead of 90% here and get ~11-5 = 6 small bets. then
161*6 = +966

I have -556-848+966 = -438, following your general method, but using ~77% as the low equity. Shoot! Looks like my 15 chip to 1 chip method needs refinement."

Well, I just gave one possible scenario, it depends on the texture of the game, what I like to call "pot shapes*", as in how many bets is likely to go in on each street, and that varies a bit from game to game, and may be greatly affected by the specific players that are live in the hand at this point. That goes for the 6*2, 4*1, 3*1, 3*1, and the 90% low equity or 77% low equity, we are both right, under our given conditions, but pot shapes can be in many forms, and the odds of facing down another A2 is also greatly affected by who your remaining opponents are and how long they will ride a naked backdoor low draw before dumping. I was going to get into starting with 3 lows, and 4 low cards too, but I thought I would wait and give you a chance to jump in here.

While there may be times where you get a free turn, and there are times where someone pops the turn, which really kills your odds, it is very situationally dependent. I just thought I would split the difference, you can revalue your hand upwards or downwards from there depending on the game texture and the tendencies of your specific opponents. I was envisioning a game where most players would abandon a pure backdoor low draw, but in some games and especially against certain opponents (some people will play an A2 with any low card on the flop, and they aint goin' nowhere). I can see how one might come up with varying low equity values here.

Thanks for all your great work. [img]/forums/images/icons/grin.gif[/img]

Don

*the more money that goes in early, compared to late I like to visualize as a downsloping "bet shape" and a very gradually inclining pot shape, and conversely, less bets in the pot early and steady bets throughout more of a flat bet shape with a steeply inclined pot shape. Of course sometimes you get heavy bets early and throughout the pot. This is of course a type of hand that likes a pot that starts out big and rises gradually at not too steep an angle. (what can I say, it's probably a little kooky)

[DPCondit]

Yes, 3.125 to 1 for a 3 card low, and 2.67 to 1 for a 4 card low. It is amazing how much extra value those extra low cards give you, you can certainly play a lot more hands in these cases, than with 5.2 to 1 odds. If I had done those calculations with my method, it would take a few extra calculations (though not with a 2 card low), so I think I'll start playing around with your method a little more. [img]/forums/images/icons/grin.gif[/img]

Now let me add this, you are using 38% for getting quartered, and 5% for getting sixthed, now even assuming none of these low draws fold on the flop (which I think is a stretch in many games, but perhaps not in others), when you have extra low cards, some of the cards that make your low will also counterfeit these draws. So that also adds quite a bit of value there. So, I think it is hard to justify 77% low equity (it should be higher), even against the most rabidly brain-dead calling stations. If you pop an ace, any two card ace-something low is dead, if you pop a deuce, you kill a 2 card ace-deuce (without backup low cards), etc. So these hands are actually stronger than you may think.

Of course, if you have A23, and you make your low by spiking a deuce on the river, you would have to consider how many people would actually be in the pot here with an A-3. Not too many fools would play an A-3 backdoor low draw (without a 2 on the flop), unless they had some part of the high equity. It adds another variable of uncertainty, but still your low equity has got to be higher than 77%. Unless you are regularly getting 5 or 6 hands on the river, I think your % equity in any backdoor low will always be well higher than when you get 2 low cards on the flop (except against certain opponents that you should usually be able to easily identify, and that applies mainly to specific people that always get married to A-2 when any low card flops, and all you have is A-2).

Just an observation.

Thanks for all your great work,
Don

[Buzz]

Don & Dave - Here’s my latest thinking on the matter. When you’re deciding whether to call a bet or not in Omaha-8, an important consideration is the odds the pot is giving you. Drawing for runner-runner low is usually a poor play because (1) it's a low percentage play, with odds of 5.2 to 1 against making low, (2) you’re only playing for half the pot and (3) there is an appreciable risk of getting quartered (or sixthed) by an opponent (or two) also making the same low.

Taking these factors into consideration, in order to play strictly for runner-runner low after a LHH flop, as a general guideline, I’m making my pot/bet requirement on the second betting round 17/1 when I hold the nut low draw. (Without the nut low draw here I’m not drawing strictly for runner-runner low at all). With the nut low draw plus excellent counterfeit protection (eg. A24K after a flop of 3JJ), I’m making my guideline pot/bet requirement on the second betting round 11/1.

That’s as close as I want to cut it. One would still have to use some sense drawing strictly for runner-runner low because a raising war on the turn is possible. For example, with a flop of 5h9dTc and a turn of 8s, two opponents, each with QJXX, would likely be jamming. Looking ahead, if I could see this happening, I wouldn’t want to take a chance of getting caught in the cross-fire. However, I believe there is money to be made here by playing the 17/1 pot/bet ratio sensibly.

When it is your turn to act on the second betting round, if the flop is has one low card and two unpaired high cards, (eg. 39Q) there have to be 19 small bets already in the pot for every small bet it costs you in order for you to play strictly for runner-runner low. But if you hold two low cards and two high cards, unless the board is paired, you almost certainly would have some sort of high draw, probably somewhat modifying that pot odds requirement, perhaps to something like 17/1. That’s assuming you’re not playing heads-up after the second betting round. Heads up there would have to be 21 small bets in the pot in order for you to play strictly for runner-runner low, but heads-up there are other considerations. After a flop of 39Q, my normal opponents would probably find some way to continue with A2XX, and if they did, my low equity would be closer to 0.77 than 0.90.

When it is your turn to act on the second betting round, if the flop has one low card and has a pair (eg. 5JJ), there have to be 17 small bets already in the pot for every small bet it costs you in order for you to play strictly for runner-runner low. Same deal heads up as above. After a flop of 5JJ, my normal opponents would probably fold A2XX, and if they did, my low equity would be closer to 0.90 than 0.77.

Facing only one small bet with 17 small bets already in the pot is not likely, but is possible in some games where there is a lot of pre-flop action. If someone capped or even three-bet the 1st betting round and if there were enough callers (five or six), and then if there were no raises on the second betting round, you would have favorable odds to call a single bet and play for runner-runner low if you had the nut low draw. After a capped first betting round with six participants, and then a subdued second betting round with everybody calling the initial second round bet, not unheard of in some Los Angeles area games, you'd actually be getting pot odds of 29 to 1. Those pot odds would certainly merit a call, even though you're only playing for half the pot, even though you only have a 0.16 probability of hitting your runner-runner draw, and even though there is a fair chance of getting quartered or sixthed. Of course if anyone raises on the second betting round, then you don't have odds to continue play when you're only drawing for runner-runner low.

When it is your turn to act on the second betting round, if you hold either A23K or A234, and if the flop is has one low card (but not A, 2, or 3), and two unpaired high cards, I think you can relax the 17/1 bet requirement to 11/1. Interesting that A234 has a better chance to make low than A23K, but not enough to matter much here.

Just my opinion.

Buzz

[beernutz]

Thanks for the analysis Buzz. I also fudge factor in the chance that I can get not only a runner runner low but a possible wheel and sweep when making the decision about whether to continue after the flop. If I'm holding A245 for example and the flop comes KQ3 then I'm almost certainly going to see that fourth card unless the pot is jammed before it gets to me. Do you think this is a case of wishful thinking?

[DPCondit]

Of course it depends on how many bets are in the pot. In a 9 person game, in a simulation, the high value of A245 rainbow, on a KQ3 rainbow flop will add about 2 1/4% pot equity. Of course that goes out the window if you add suits. If we had, say 13.5% equity in the pot, it is now 15.75%. If we were to use Buzz' rule of thumb for 11 to 1 pot odds, then you could probably go 9 1/2 to 1.

Don

[DPCondit]

Small quibble here.

Generally Buzz, your math is excellent, but there was a tiny slip up in case 3. I show the odds should be about 3.024 to 1, not 2.67 to 1. It appears you duplicated 47 and 48. I know it's easy to make a little mistake when you are putting that many calculations into one post.

Don