Okay, so I'm pretty comfortable with mathematics in general but some parts of probability theory trip me up.

An example:

Hero holds: AQo

Flop: A29 rainbow.

For argument's sake let's say that there is one villain and that we have a very solid read on him. Based on preflop and flop action, we have decided that his range is QQ, KK, AA, AK. Further we consider each of these four holdings to be equally likely.

Turn brings another Ace.

Now, before any action on the turn, we can obviously re-evaluate the probability of villain holding AA to zero.

My question is: do we also reevaluate the likelihood of him holding AK (based on the fact that there is now only one A left for him to potentially hold)? or, assuming we still trust our earlier read, should we consider his range to be: QQ - 33%, KK - 33%, AK - 33%?

On the flop:
AA = 1 combo = 1/18 = 5.6%
KK = 6 combos = 6/18 = 33.3%
QQ = 3 combos = 3/18 = 16.7%
AK = 8 combos = 8/18 = 44.4%

On the turn:
AA = 0 combos = 0/13 = 0.0%
KK = 6 combos = 6/13 = 46.2%
QQ = 3 combos = 3/13 = 23.1%
AK = 4 combos = 4/13 = 30.8%

AA became impossible, and AK became less likely because it now accounts for fewer of the possible combinations given your specific read.

EvilSteve,

whilst your analysis is obviously correct, strictly speaking it does not answer the question posed by OP, who stated that the holdings AA, KK, QQ and AK were equally likely on the flop based on the action and reads so far. I read that as villain has each of the holdings 25% of the time. This could happen if hero has a strong read that villain uses a mixed strategy with these holdings with weighting percentages leading to 25% for each.

Assuming that this is what OP meant (far from sure here, OP please confirm), the results are modified to

AA 25% --> 0%
AK 25% --> 1/6 = 16.67%
KK 25% --> 5/12 = 41.67%
QQ 25% --> 5/12 = 41.67%

You forget he's holding a Queen, so only 3 QQ combos.

takovich:

that is what I meant, but I only really posed it that way for maximum simplicity. Evilsteve answered the somewhat more complex problem that this one is a subset of.

more specifically what i was asking was: when a community card comes that represents part of an opponent's range, does that part of his range become less likely?

and the answer is yes.


Thanks to both of you.

No, I did not forget that.