In the post correcting errors, I went over the assumption of what hand people should play to have a 50% chance of having the best hand. I will use this assumption to determine GIVEN a fold how does that impact the probability of a given card in the deck. We talked about how a better model means better results, now it's time to build one.
At 9 handed table, first to act (UTG) should play the top 8.3% of hands to have a 50% chance of having the best hand. At this point we are making an assumption of not only the range BUT what hands he will play.
88+, AJ+, ATs+, KTs+, QJs
There is an outside chance that he will not play KT+ and QJs and AJ and instead play A2s+, or suited connectors or some other strange combination. He also could play a much wider, or much tighter range. But because of the "outside chance" we will reduce all cards in the deck weighted slightly by higher but we will do this later. For now let's just look at the percentage of hands that this person folds that contains one of the following:
A, K, Q, J, T, 9, 8, 7, 6, 5, 4, 3, 2
and the percentage that contain TWO of each of those cards.
Probability opponent has folded one:
ace 0.115131579
king 0.134868421
queen 0.138157895
jack 0.138157895
ten 0.15131579
9 0.157894737
8 0.157894737
7 0.157894737
6 0.157894737
5 0.157894737
4 0.157894737
3 0.157894737
2 0.157894737
probability opponent folds 2
8+ 0.000000000 (we will need to adjust this upward slightly)
7 0.004934211
6 0.004934211
5 0.004934211
4 0.004934211
3 0.004934211
2 0.004934211
ok.Now. Probability opponent did NOT fold one:
ace 0.884868421
king 0.865131579
queen 0.861842105
jack 0.861842105
ten 0.84868421
9 0.842105263
8 0.842105263
7 0.842105263
6 0.842105263
5 0.842105263
4 0.842105263
3 0.842105263
2 0.842105263
Probability opponent did NOT fold 2:
8+ 1.000000000
7 0.995065789
6 0.995065789
5 0.995065789
4 0.995065789
3 0.995065789
2 0.995065789
As such you can determine probability that you have X cards in the deck after one fold.
To do this you must take the probability you did NOT fold 1 of a given card TIMES 4 (the number you have of that card if there is no fold). PLUS 3 times the probability you DID fold exactly 1 hand.
Then you have to take probability that you did not fold 2 of a given card times 4 PLUS the probability you folded exactly 2 TIMES 2 and subtract that number from the other total. For example:
probability that you don't fold a 7 = 0.8421053 times 4 = 3.368421053 probability you fold a 7 0.1578947 times 3= 0.47368421.
probability you fold 2 7s 0.0049342*2=0.009868421. Probability you don't = 0.995065789475981*4= 3.980263158 +0.009868421=3.99013157895196 - 4 =-0.009868421
3.368421053+0.47368421-0.009868421=3.832236842 effective cards in the deck.
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
ok. This knowledge could actually change the optimal behavior of the 2nd person to act. Now what we really want to do at this point is determined based upon a conditional deck what hand range has a 50% probability of being best. Ideally, you actually want to determine the probability that the hand has positive expected value and if it's 50% or greater than play the hand.
This requires more work. For now we will have to come back to this problem. Afterall, the next player to act should have a slightly different set of assumptions given this effective remaining deck.
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