Freddy limps with Ac As, Daniel raises around back with KhJs... Freddy reraises to about $30k (around 3x Daniel's original raise). Freddy had $80k behind it so his implied odds are about 2.5:1. Daniel calls.

Board comes 9cJcTh... pot is $75.3k

DN checks, FD goes all in for $80k... DN sorta nonchalantly calls saying he is getting about 2:1 and he thinks he is a 2:1 dog.

I would be interested to hear his thinking on this hand---especially when he asked to run it twice as a 2:1 dog.

Also, as a more general comment---and please don't take this as a shot at any of the pros---it just seems that a lot of the big losses these guys get themselves into could be avoided simply by better preflop selection. Rarely do they scoop a pot by raising late with junk and then betting the flop. These guys just don't let that happen very often. More likely, one is going to flop a piece and then not get away from it. This seems like poor play. I would like to understand it some from the perspective of the pro themselves.

Well, I'm not DN, but he's got nine outs here if he put Freddy on AA.

4 Queens - for an inside str8
2 Jacks - for trip jacks
3 Kings - for 2 pair

Nine outs with 2 cards to come is approx 36% (9 * 4 using the rule of 2/4), which is about 2-1 odds.

Not sure why he asked to run it twice though.

Wouldnt you want to run it twice being a 2:1 dog? 2 chances of splitting the pot against sudden death.

Running it multiple times does not reduce your equity. He's a 2-1 dog once the money goes in whether they run it once or 75 times. Dannys not putting up more money once he finds out he is a dog.

Running it more than once reduces variance, why would anyone want to run it multiple times as a dog if it took money out of their pockets? These guys are at the top in the game of poker and I'm pretty sure they know what they are doing.

Right. Duh. Brain fart.

So I guess then my question is as a 2-1 favorite, why would Freddy agree to this?

Just to reduce variance?

overall, probably. maybe there will be a time when he gets his money in against DN as a 2-1 dog. will DN now be more or less likely to agree to run it twice?

I'll be glad to run it as many times as you like when I'm the 2:1 favorite.

First off let me say I am not clear exactly the way running it twice works. Hopefully Daniel will read this thread and comment on the following analysis.

Now, Daniel is about a 2:1 underdog on the flop when the money goes in. To simplify matters let's look at the following approximate probabilities (I realize that if DN hits one or more of his outs in the first run then there are fewer outs for him to hit on the second run, but for now---to simplify things---we will ignore this subtlety):

Daniel goes Win-Win: 1/3 * 1/3 = 1/9
Daniel goes Win-Lose: 1/3 * 2/3 = 2/9
Daniel goes Lose-Win: 2/3 * 1/3 = 2/9
Daniel goes Lose-Lose: 2/3 * 2/3 = 4/9



THEORY A: SPLITTING THE EXISTING POT BETWEEN THE RUNS

The pot is fixed once the money goes in (in this case it was $235.3k). Since the money went in after the flop was shown, only the turn and river cards are run twice. The winner of each run wins half the existing pot ($117.65k).

Daniel goes Win-Win: 1/3 * 1/3 = 1/9 for $235.3k - $110k = +$125.3k
Daniel goes Win-Lose: 1/3 * 2/3 = 2/9 for $117.65k - $110k = +$7.65k
Daniel goes Lose-Win: 2/3 * 1/3 = 2/9 for $117.65k - $110k = +$7.65k
Daniel goes Lose-Lose: 2/3 * 2/3 = 4/9 for -$110k

Daniel's expectation: (1/9)($125.3k)+(4/9)($7.65k)+(4/9)(-$110k) = -$31.57k

(For the record, this is what I believe they are doing.)

THEORY B: PAYING THE ALL IN AMOUNT FOR EACH RUN

For the all in call, Daniel called $80k into a $155.3k pot. Here we will act as if each player will basically play the all in betting sequence twice with the same given flop. The pot then becomes $75.3k (preflop) + 4*$80k = $395.3k and each player wins half of the (new) TOTAL pot.

Daniel goes Win-Win: 1/3 * 1/3 = 1/9 for $395.3k - $190k = +$205.3k
Daniel goes Win-Lose: 1/3 * 2/3 = 2/9 for $197.65k - $190k = +$7.65k
Daniel goes Lose-Win: 2/3 * 1/3 = 2/9 for $197.65k - $190k = +$7.65k
Daniel goes Lose-Lose: 2/3 * 2/3 = 4/9 for -$190k

Daniel's expectation: (1/9)($125.3k)+(4/9)($7.65k)+(4/9)(-$110k) = -$58.23k

(I don't see how this is what they are doing since if one of the players is "all in", where do they get the money to "pay" for the second run?)

WHAT ABOUT JUST RUNNING IT ONCE?!

Daniel's expectation: (1/3)($125.3k)+(2/3)(-$110k) = -$31.57k

While this may surprise some, it doesn't surprise Daniel or any of the other pros! The reason they run it twice is their EV does not change---but running it successively does decrease the variance.

I would ALWAYS run it twice as a 2:1 fav!

I don't see why (other than pure gambling reasons) someone would choose to run it more than once as a 2:1 dog.

Maybe im being stupid,but lets use a more obvious example. Barry against Farha. When Farha hit his set against the kings, he offered to run it again. Barry turned him down, maybe because he said no pre-flop, maybe because he wasnt sure Farha had the set as he hadnt turned his pocket cards over. Wouldnt you jump at the chance of having two shots at making your hand? four chances of hitting an ace (turn + river twice,as opposed to once)?

As outlined above, running it as many times as you want does NOT affect the equity, just the variance. Someone with a big fav should crave fixed equity and reduced variance. On the other hand, a big dog should crave higher variance and hence should only want it run once!

I still wanna know exactly how they do the payoffs when they run it twice, whether it is Theory A or Theory B above. If it is A as I suspect, it really just doesn't matter whether they run it one time or n times even if n is very large.

Obviously im only talking about variance.Obviously they only play for the money in the middle no matter how many times they run it, not multiples.Would you rather have four chances of hitting your hand or two?So BG should see his opponent with a set and 'crave higher variance' knowing that it is harder to hit an ace with two attemps than four (8/1 against 4/1 approx?)

It seemed to me that from all of the episodes so far, Daniel was happier to run it twice when he was the dog, but only wanted to run it once when he was the fav.

Theres variance and there's luck. obviously over say 50 runs,. 100 runs variance will work itself out. only running a hand 2-3 times isnt going to affect variance, but you can get always get "unlucky".

Luck is not a scientific term and as such I do not know what it means.

As for variance, it does not take a large number of trials to shrink the variance. Two is enough to have an effect although a larger number would certainly decrease the variance even more; that is why the pros care enough to run it twice.



You must keep in mind that if they agree to run it n times and you (as the underdog) spike m of those times (where m is certainly going to be much smaller than n in general), you only get m/n of the total pot.

It is NOT "if I spike my dog once I win the whole pot". That would be an absurd transaction for the favorite to engage in.

At the end of the day, if BG gets to run it til the deck is empty in the dealers hand it will NOT affect his expected value. I think you are thinking that it does (based on your last sentence).

Of course it does, any action that increases his chance of splitting the pot increases his expected value. If i placed a bucket 20 yards away and said i will give you $20 if you can throw a coin into it. Do you want one attempt or five?If BG gets to run the deck, he will hit his ace and split the pot. the pot isnt divided into how many times its run. In the same way if its run three times it isnt split into thirds-they sometimes do this because they want an outright winner i.e 2-1 or 2-0. Winner takes all.

You are just plain wrong because you are thinking that if you spike once as the dog you get the WHOLE pot. YOU DO NOT. Why would ANYONE who is the favorite go for such a deal? The dog only gets the ratio m/n of the pot where m is the number of times he spikes and n is the total number of runs.

The math is simple. It does NOT change the EV with the way they play it (which is NOT the way you are offering to play the game).

Look a few posts up at my THEORY A calculation and my WHY NOT JUST RUN IT ONCE calculation. That should convince you.

where do i say they get the whole Pot?- SPLIT THE POT.If your ahead as far as Farha was, why give variance the opportunity of kicking him in the ***? variance can only work against you. and if you read what i said about running it three times your m/n ratio doesn't work either.THERE IS A CLEAR WINNER.......!"!!"! . get it? Winner takes all not 2/3rds.

I am sorry but you just don't understand how running it twice works. The winner of run #1 gets 1/2 of the total pot. The winner of run #2 gets 1/2 of the total pot.

If you ran it three times the pot would be in thirds and the winner of each run would get 1/3 of the total pot.

If you ran it n times, the winner of each run would get 1/n of the total pot.

Ergo, if you spike on him m times (which IS going to be small since you are the 2:1) dog, you only get m/n of the pot.

Expected value is absolutely unchanged no matter how many times you run it.

If you are still not convinced, read Daniel's own reply in this thread:

No you are making a mistake. Run three times winner takes all. Your not reading what i say, plus your making assumptions. I surrender..good luck

LOL... How do you run an event and winner take all if there are different outcomes in each trial?

Andy Beal is right. But can't anyone read the sticky about not posting your hands here?

This is not a hand of mine. It is a hand of Daniel's.

First off, glad to see you posting here Andy!

I have to agree with Andy, and Theory A looks correct.

However, whether you are the fav or dog, variance is variance, and less of it is usually beneficial to both parties. In the long run, the EV will catch up anyway, so why not make the ride less bumpy?

Phlat_________

Well one might turn around and argue that variance is the friend of the fav since reducing variance just makes higher probability events more "likely" to occur on this small set of trials.

On the other hand, one might very well argue that if you are the dog, then you NEED variance to help your low probability event "occur" and therefore should want to just run it once.

However, I think human nature takes over whether you are the dog or the fav: both want a better chance of losing less even if it means sacrificing some of their chance to win more. I talked more about this here (see the 3/22, 9:06 pm post of mine).

Winner takes all if he wins all three, so you are correct if I assume you left something out of your post. If I don't assume that you left something out of your post, then I know you're wrong.

Just re-read what Andy wrote and grasp why he is correct.

Good day.

True, and I see your point. By the way, very nice theory post.

I still maintain my original position. Variance is a two way street that really only comes into play in the short term. Bank roll determines the "ride" you may want to take.

I agree with your first paragraph, with "small set of trials" being key here. However, in the second paragraph, I don't see how variance will help your low probibility event occur. Again, variance is a two way street. The positive swing will equal the negative swing and -EV will emerge in the end. Win less/lose less, or win more/lose more. Are you saying that when you are +EV you are willing to gamble less, and when you are -EV you are willing to gamble more? I think this is the true "human nature" of a poker player.

I think the whole running it X times is probably more for pro's entertainment and bankroll security. Personally, whether I am + or - EV, I will take reduced variance any day of the week.

Phlat_________

i would run it twice as a 2:1 dog. I get to see 4 cards and only one of those has to be a face card for a split. if you get a face card on both runs then you take it down

Alright, my first post might as well be a stupid question.

I don't feel like doing the math so I'll ask one of you. If the deck is not reshuffled and it's a situation where someone is drawing to a one-outer doesn't that effect the EV since their is no possible way to scoop the pot?

Inuitively it would seem so, but what was proven in the thread:

http://www.fullcontactpoker.com/poker-foru...showtopic=54819

that in all situations, even without shuffling, the EV is always the same whether it is run once, twice or even n times. Ah, the power of proof.

But, if you actually do the calculations, you will find the EVs are the same even with only 1 out. Might be counterintuitive, but true nonetheless.

My calculations show

EV(once)=.089W-1.91L
EV(twice)=.089W-1.91L

Here I have assumed 45 unknown cards and 1 out. W is the amount won per run, L the amount lost per run. (Treat running it once as running it twice but act like the same cards that fell the first time fell the second time.)

Not necessarily.

I think what you're getting at is the notion of convex preferences. Thing is - they're addicted gamblers (they arent 'problem' gamblers until they start losing ) - they enjoy gambling for the sake of gambling.

While each dollar is of diminishing marginal value in and of itself (implying they ought to prefer averages to extremes), the bet itself isnt significant relative to their overall wealth (at least int he case of daniel). There is some marginal net loss, but when you're operating so far along your marginal utility curve, it's pretty inconsequential. If it's not entirely clear, i can draw out a pretty diagram in paint. If joe blow sat at the table with his life savings, he would clearly prefer the hand be run more than once. In fact, he'd probably want it run an infinite times - or in other words, that he be paid precisely his expected value. Why? Because he prefers averages to extremes, because the marginal value of each dollar declines. The first $10,000 is more valuable than the next, which is more valuable than the next. And so on - but this fall off comes at a declining rate. Even if he LOVES gambling, he's not willing to screw up his life that much to get some action. But if he was a billionaire, he'd be so close to indifferent to the risk involved that, if he enjoyed gambling at all, he wouldnt care about running it multiple times.

More significant for "the pro's" is the joy they get from gambling. For whatever reason, they get off on having their money arbitrarily redistributed between each other. Mmmm... arbitrary redistribution.

Hate to say it Davezz5, but Andy is correct. I understand that it seems a bit illogical because you are looking at it from the perspective of having more 'chances' to hit your outs. But trust us, the math works out even.

And if you don't believe that, think of it from the favorites standpoint. Why would any player give you more chances to catch your outs if it was going to cost him money in the long run? Winning over time is why we play poker in the first place.

Mark

I'm not sure what your point was, but the math says running once, twice, 3 times, or thru the deck all have the same EV. Period. It is a mathematical fact that whether one understands it or not or believes it or not is just as true as the Laws of Thermodynamics or gravitation. Similarly, the math indisputably says that the variance is lowered each run.

There can be speculation on whether a certain player would want fixed EV with higher or lower variance when he is the dog/fav, but those are probably personal preferences based on many factors.

In a nutshell, the amounts of money these guys are playing for ARE meaningful (even to them). If they were not, they would increase the stakes until they were because the game wouldn't be much fun otherwise.



Exactly.

Huh?

What did you mean by,

"However, I think human nature takes over whether you are the dog or the fav: both want a better chance of losing less even if it means sacrificing some of their chance to win more. I talked more about this here (see the 3/22, 9:06 pm post of mine).

?

That's what i was responding to.

After a lot of words in your post (convex preferences, etc.) you say that the pros run it twice "because of their love of gambling".

Running it twice reduces variance and builds a hedge against losing the most.

That doesn't sound like someone "gambling for the fun of it"---quite the opposite. It sounds to me like someone who is trying to minimize his loses. And that's all I was saying in the first place.

I didnt read all of the posts but when they ask to run it twice it means if they win 1 out of 2 they split the pot, if they win both they get the full pot.

I said the exact opposite.

I said joe blow would prefer it be run an infinite number of times.

Joe blow is no pro.

He's an example of a poor shlub who, for whatever reason, got involved in a high stakes game of poker and was faced with the decision being discussed.


Pro's (or at least a good portion of them) probably wouldnt care, and would just wnat to run it once if it was for a negligable amount relative to what they have. Either to save time, to save face and not be seen as 'scared', or because they genuinely enjoy gambling.... a hobby that, for many, is what drew them to playing cards in the first place.

Hard to follow your logic or point here since the very first post in this whole thread is about two pros that chose to run it twice.

Clearly it wasnt negligable to them.

I was responding to the statement i quoted, not the original post.

Nevermind.

Sorry to be redundant and quote myself but I would really like to hear Andy's thoughts on this.

Andy, it sounds like you are saying it is more advantageous for a player with a -EV hand to run it multiple times. I still fail to see how this is true. Advantage is measured in terms of EV. Variance is measured in terms of trials. As you know, running it many times decreases variance, and running it fewer times increases variance. In no way does the Variance/Trials outcome have any effect on the Advantage/EV. Bankroll management and entertainment value are the only considerations when it comes to running it N times.

I think Abbaddabba was only stating why some people would choose less/more variance by running it once/multiple time(s). ie. someone with life savings on the line wants less variance, whereas a seasoned pro could live with more variance. A dollar means more to some than others. But this does not change our EV/Variance debate.

Andy, I think you have fallen into the gamble more to catch up when -EV and preserve bankroll mode when +EV mindset. I honestly don't think there is any tactical advantage there.

Phlat___________
PS. LOL, this is too funny. "Joe blow is no pro."

I can see how you might infer that but let's be careful here: I offered a conjecture on why I thought a dog might want to run it twice, not a statement that running it twice is advantageous for the dog.

The reason I belabor this last statement especially is because the math clearly says that it is really neither advantageous or disadvantageous---the EV's are equal whether you run it once or more. If you are an 11:1 dog, your opponent is most likely gonna scoop the whole thing if you run it once or twice or N times. On the other hand if the pot is $250K+ like on High Stakes Poker on GSN and Esfandiari turns over AA while Elezra turns over JJ, Elezra may very well choose to run it twice to give him a better chance at losing less. It just depends on the risk/reward characteristics of the individual players involved I think. Similarly if you and I are playing for a ton of money and you turn over AK to my QQ we might (implicitly) decide to "call it a tie" for all that money by running it twice since the chances are very good that we will split.




I'm not sure what you mean. I don't think I made any assertions in this direction. Because if I did, I would have implored the dog never to run it twice since it is much more difficult to strike the big payday a gambler would seek!

Above was your assertion in the direction I was talking about when I said "Andy, I think you have fallen into the gamble more to catch up when -EV and preserve bankroll mode when +EV mindset." Gamble more here means running once when -EV.

Increased variance does not help a low probability event occur, it only increases the range of possible outcomes. Over a large sample the -EV will come out anyway.




Becareful with your choice of words here, as Elezra does NOT have a better chance at losing less. He is simply limiting BOTH winning and losing potential, ie. variance. I agree that these considerations depend heavily on the risk tolerance of the individual player.

Phlat__________

OH, again let me be clear. I wasn't saying that high variance "helps" a low probability event occur, I was saying one might try to argue that. Of course there is nothing one can do to hurt OR help a specific probablity event occur any more than one can affect the gravitational constant of the Earth.

No he does have a better chance of losing less by running it 2x (of course, he has a worse chance of winning more too). It's basically the same caluclations that I already did for the Deeb-Negreanu hand here, post #7. If they run it once, DN has a 67% chance of losing $110k and 33% chance of winning $125.3k. If they run it twice, Daniel has a 44.4% chance of losing $110k, a 44.4% chance of chopping the pot (for a modest win of $7.65k due to the dead money in the pot), and only an 11.1% chance of winning $125.3k. So by running it twice, Daniel has a considerably better chance of NOT losing the max of $125.3k.

In a situation where the EV is identical (as running it twice always is), I think the only variables in play are metagame effects of variance: (1) whether or not, if you bust the other player, they will get up and leave, and (2) whether or not, if you give them a beat, they will tilt.

Say you've got a strong player sitting at the table that is unlikely to rebuy if they go bust, with a weaker player waiting in the wings. This pushes the advantage to running it once (whether dog or favorite), because of the chance to bust the player and lower the overall skill level of the table. Conversely, busting a weak player with a strong player waiting decreases your overall EV by making the table stronger. This situation gives value to running it twice, whether dog or favorite; keep the table weak, and increase your chances of getting your money in with better EV in future hands.

On the psychological level, I can make an argument, I think, for running it only once as the dog in a situation where, if you lay a beat on a player for their whole stack, they will rebuy, go on tilt, and drastically increase your EV in the near future. In another hand, on an earlier night, DN was willing to call in what he knew was a very negative EV situation in order to put someone (I forget who off the top of my head) on tilt. While that's something I don't think I would ever do (and I don't agree with the play under the circumstances for sure), I see the logic behind it. In a situation that doesn't affect your EV, such as running it twice, I think you want to do whatever you can to bump up the variance if you feel you have a better psychological capacity for handling the swings than your opponent does.

I think DN and FD's mutual decision to run it twice reflected their natural desire to lower variance as well as (possibly) a mutual respect insofar as neither one felt they would be likely to tilt the other. Maybe Freddy also felt like he had an advantage over DN so long as DN was playing hyper-aggressive, but felt that if he took 100k from him, Daniel would calm down, play tighter, and lower Freddy's EV - kind of an inverse tilt sort of thing.

Who knows really? When EV is not affected, very small considerations can push the balance one way or another.

first of all, I havent read all the replies, I just found 'Andy's' calculation is logically unjustified. Based on that, this can't be the Andy Beal who is a very good mathematician.

Now, why did I say that?

There is one fact, that 'Andy' ignored, and he called it 'subtlety', that there is only 52 card in a deck. Personally, I don't think this is subtle at all, nor it is a minor issue. In fact, this is the ONLY issue that matters.

Ignoring this fact, which means the probability of drawing out as a underdog remains the same, the equity will OF COURSE, ALWAYS stays the same. You really dont need any calculation to draw that conclusion. Because all you did, is run the EXACTLY same hand a number of times, and each and everyone of them has the EXACTLY same probabilty of winning/losing.

Now why isnt it good to run it more times when you are a FAVOURITE?

NO, IT IS NOT THE VARIANCE

That is because, you are effectively giving more chances to get out DRAWN.

REMEMBER there are only 52 cards in a deck, and are only a few cards that can help the DOG. As a favourite, you DONT WANT TO SEE THEM. If you ran it twice, you are given out more chances to SEE them, that is IT.

The chances of hitting 3 cards when there is 35 cards left is much higher than there is 40 left, isn't that obvious?


I think this is very simple to understand?

Now because this is a 'mathematical' discussion, let's see some numbers.
Lets use a rather simple example (coz I am lazy)

AK vs KJ, and you are dead to a JACK on the flop:

you have two chances to hit remaining 3 jacks in the deck, the estimated probability is: (considering only you and your opponent's hands)

(3/45)*(39/44) = 5.9%
-> one jack on the turn, a card that is NOT an ACE on the river (Situation A)

(42/45)*(3/44) = 6.36%
-> one card that is NOT an ace on the turn, and a jack on the river (Situation

(3/45)*(2/44) = 0.3%
-> hit runner runner Jacks (situation C)

all toghether the estimated probability that KJ will draw out on AK is:
12.56% (might be a different from the simulation results, this is just an estimate anyway)

So the probability of NOT drawn out is about 100% - 12.56% = 87.44% (Situation D)

Now what if we run it twice?

The first run obviously is identical to the above

now the second run:

provide that it is situation (D)

we had a new set of probabilities, similar to the above caculation only now there are only 43 cards left:

(3/43)*(37/42) + (40/43)*(3/42) + (3/43)*(2/42) = 13.12% (E)

Notice 13.12% > 12.56%

This is the probability that KJ drawn out on AK PROVIDE THAT it didnt drawn out in the first run, the overall probability for this to happen is:

P(D) * P(E) = 87.44%*13.12% = 11.47%

THIS IS A SPLIT POT, but this is only one situation.

IN total we have the following situations:

1) when we draw out on AK at the one time, but didn't draw out at the other time (this has eight combinations, 1st turn, 1st river, 2nd turn, 2nd river, 1st turn and river and NO JACKS in 2nd, 1st turn and river and one ace one jack, 2nd turn and river and no jacks in 1st run, 2nd turn and river with one ace and one jack in 1st run)


2) when we draw out on AK at the first time AND we draw out on AK at the second time (this has eight combinations, 1st turn+2nd turn, 1st turn+2nd river, 1st river+ 2nd turn, 1st river + 2nd river, 1st turn and river + 2nd turn and no ace on the river, 1st turn and river + 2nd river and no ace on the turn, 1st turn and no ace on the river + 2nd turn and river, 1st river and no ace on the turn + 2nd turn and river)

3) we didnt draw out on neither runs:
this is rather easy to calculate: 87.44%*(1-13.12%) = 75.98%

Notice the drop in probability compare to situation D

which means what? your AK is that 87.44% - 75.98% = 11.46% less likely to hold for the whole pot!

Since no one is paying me for doing this, Im not gonna bother about the details. But HOPEFULLY you can see why it is NOT GOOD to run more than once as a favourite.

I don't understand what all the math is supposed to be about here. The EV of running it twice is the same whether you run it once, twice, or eight times. The only thing you affect is the variance. In effect, you are putting yourself in an identical situation twice for half the stakes each time. Yes, if you are the favorite, you gain the additional chance of your opponent getting half the pot, but you also essentially kill your opponent's chance of scooping it with a bad beat. The only variable that is affected is your variance.

READ my post carefully, your assumption is WRONG

THERE ARE ONLY 52 CARDS IN A DECK and it matters GREATLY

that is, if you didnt get the card you want from the 1st run, you have a HIGHER probability of hitting it at the 2nd run. If you run it a 3rd time, the probability is going to be even higher. Understood?

if you cant understand the math, at least try to understand the logic

As I mentioned in the other thread, your logic is flawed.

If you don't hit on the first run, you have a greater chance to hit on the second, but if you do hit on the first you have a lesser chance on the second. These two effects cancel each other out.

The chance of only getting half is mitigated by the reduced chance of leaving with nothing, both for the favourite and for the underdog.

Ah, I love being talked down to by someone that I (hope and pray) has never taken a statistics course. Mriya, you are simply wrong. Not "I disagree" wrong, but "2+2=5" wrong.

Let's boil it down to the basics. Let's say you have your opponent drawing to 1 out with 1 card to come (under-set vs overset, for example). There's an even $1000 in the pot. With 4 cards on the board and 2 in each of your hands, that leaves 44 cards unaccounted for. You are a 43:1 favorite; your opponent has a 1 in 44 chance of sucking out.

Therefore, running it once, your expected value is (43/44) * 1000 = $977.27.

Running it twice isn't much more complicated. On the first card, your EV is calculated in exactly the same way: your chance of winning the first half of the pot is 43/44, just as it was running it once. Therefore, on the first run, your EV is (43/44) * 500 = $488.64

On the second run, 1 out of every 44 times your opponent will have sucked out on the first run. When this happens, your opponent will have no outs left and be drawing dead, so your EV for this happening is (1/44)*(1/1)*500 = $11.36.

The other 43/44 times, your opponent will not have sucked out on you and will be drawing live, with 1 out out of 43 cards. Therefore your EV in this instance is (43/44) * (42/43)*500 = $477.27.

To get your total EV for running it twice you just add your three EV's together: $488.64 + $11.36 + $477.27 = (gasp!) $977.27, the exact EV of running it once.

I've expanded this way more than it needs to be, with the hopes that you'll be able to follow and see why running it twice does not raise or lower your EV in any way at all. Your variance is reduced, but that's a complication more complex than you're probably ready for at this point.

Burnnnnnnnnned! You're the insult master! (Aqua Teen Hunger Force quote)

And you're right...nice example too.