Then you do this for all of them and get X cards of given value in the deck:
ace 3.884868421
king 3.865131579
queen 3.861842105
jack 3.861842105
ten 3.84868421
9 3.842105263
8 3.842105263
7 3.832236842
6 3.832236842
5 3.832236842
4 3.832236842
3 3.832236842
2 3.832236842
Today we are going to adjust for the probability of each particular hand and then come up with an adjusted hand ranking chart. Since an ace in the deck is most likely to remain, there will be an increased chance of being dealt aces As such, the hand ranking must be based upon the probability of having a better hand rank than remaining opponents.
First lets see how you would typically calculate the chances of getting
1)AA,
2)AKs
3)AKo
4)AK
1)4 cards out of 52 in the deck times 3 cards of that kind remain out of 51 remaining. (4/52)*(1/51)=~0.00452489
2)AKs is the same as KAs and thus we must add the odds of being dealt any king then a matching suited ace that is suited to the odds of being dealt any ace then a matching suited king. As such ((4/52)*(1/51))+((4/52)*(1/51))=~0.003016591
3)AK offsuit - Same principal of AKo or KAo except we cannot include suited so there are 4 aces then only 3 nonsuited kings OR 4 kings then only 3 nonsuited aces
=((4/52)*(3/51))+((4/52)*(3/51))=~0.009049774
4)AK- This includes AKs and offsuit. This time it is:
=((4/52)*(4/51))+((4/52)*(4/51))=~0.012066365
Now with an adjusted deck the odds are calculated basically the same EXCEPT we will have fractions of cards left that vary depending on which card, and it will be of a 50 card deck instead of out of 52. So now
1)AA - (3.88486842090556/50)*(2.88486842090556/49)=~0.004574422 We KNOW that there will be EXACTLY 1 less ace in the deck after you are dealt one so we can simply subtract 1. What is interesting is that although there is effectively a larger probability of a particular face card such as a jack being drawn, once it is drawn once, the odds of the 2nd being drawn are less because it is a reduced deck with MORE than 1 card missing. As such, ONLY aces has a higher probability of being drawn where all other pairs decline in likelihood given a fold.
2)AKs - (3.88486842090556/50)*((3.86513157936705*.25)/49)+(3.86513157936705/50)*((3.88486842090556*.25)/49)=~0.003064393
In this case, given that you are dealt an ace or a king there is a chance that there isn't 1 of a matching suit in the deck. removing an ACE has no influence on how many kings left either. As such I chose to take 1/4th of the cards of that kind left in the deck since if there were 4 you would only use 1, but since there is less than 4, you have to use less than 1. I cannot be sure with confidence this is 100% accurate or how it should be approached, but this is how I chose to do it.
or
3)AKo - ((3.88486842090556/50)*((3.86513157936705*0.75)/49))+((3.86513157936705/50)*((3.88486842090556*0.75)/49))=~0.00919318
Again, I decided to use 75% of the number of cards in the deck because I don't know if all suits matching the first card will be available, or if I should just subtract 1 but I chose to do it this way.
4)AK - ((3.88486842090556/50)*(3.86513157936705/49))+((3.86513157936705/50)*(3.88486842090556/49))=~0.012257574
As you may have noticed, the chances of AK increased after 1 fold by about 1.58% while the chances of AA increased by 1.1%.
This process must be repeated to develop an entire distribution of hand probability.
Now using this, we can begin to construct a chart that determines what hand range one should play with X opponents to give themselves a 50% chance of having the best hand.
I believe the "percentage" that one should play to have a 50% chance of having the best hand is still the same HOWEVER, the hands with aces in them become more and more likely for remaining opponents to have,thus a larger and larger percentage of hand rankings will then be made up of premium cards. As a result there STILL will be an adjustment on what hand range to play, but the "top x%" hand ranking will still be the same as seen below.
| SB/BB | 50.00% |
| button | 29.28% |
| cutoff | 20.65% |
| hijack | 15.90% |
| lojack | 12.95% |
| UTG+2 | 10.90% |
| UTG+1 | 9.43% |
| UTG | 8.30% |
The actual hand range after the first fold is calculated after this work and will be shown below. In the next part we will build off of the assumption of a particular hand range and determine what information two folds gives off.
UTG same as before: 88+, AJ+, ATs+, KTs+, QJs
UTG+1: Also same as before but before we had 9.35% of hands represented, this time we have 9.39% 77+, AJ+, A9s+, KTs+, QTs+ Also, because it represents slightly more hands it goes from a 50.295% chance of the best hand down to 50.11%.
UTG+2...
Now we have more information with yet another fold and 2 cards removed. We have to use this assumption yet again to adjust the information. I believe each fold will compound on the small changes and ultimately the hand range from the blinds and on the button will be very different if folded to. But perhaps the difference will end up being relatively insignificant to starting hand ranges. Eventually we will see, but right now this is too tedious to continue just yet and I feel like I may be missing something so I will have to re-evaluate after I have gotten some sleep.
Just doing some basic math assuming that each particular hand value declines by the same percentage and I calculate that AA will be 8% more likely that it is dealt to the big blind if you are in SB and everyone else has folded without action, and AK will be 13.5% more likely. However, I believe that with wider hand ranges, folds give more information. As opponent's hand range widens, it will likely include more aces. That is much lower than Barry Greenstein's simulation where he had much wider hand ranges including A2+ in late position and AT+,A2s+ in middle position and concluded that AA will be over a 0.7% chance of being dealt in BB rather than the usual 0.45%. This will be some significant information that may change when you should be willing to race with AK vs avoid it, and why you perhaps should tighten up a bit more in late position when facing a reraise than theory might suggest.
Ideally I want to try to determine an exact range based upon my assumptions of hands, and secondly perhaps some assumption of more loose play that is perhaps more similar to Barry Greenstein's work here.
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