How do you calculate the pre-flop odds of making a hand after the river card? For example, what are the odds of making a set if you hold a pocket pair after the river card? What are the odds of making a straight if you hold A3 after the river card.
Each scenario is different. Some are straightforward (or at least feasible) to calculate mathematically. For those that are not, it is simplest to iterate all hand combinations (using a computer) since the total number of combinations is not that large.
For your first example, if you hold a pocket pair then there are two cards (out of the remaining 50) that will make you a set. To not make a set, you'd need to avoid these two cards five times in a row, for which the odds are:
(48/50) * (47/49) * (46/48) * (45/47) * (44/46)
which is 80.82%. So the odds that you do make a set by the river are the remaining 19.18%.
Note that this 19.18% includes the possibility of making four-of-a-kind using your pocket pair or a full house which include your pocket pair as part of a 'set'. Both the 80.82% and the 19.18% also include (for example) straights and flushes. So if you mean 'a set or better', or 'a set but nothing more', then those are different questions and much more complicated to calculate.
The question of making a straight with A3 is also harder. You need to consider 5-high, 6-high and 7-high straights, as well as Ace-high straights. What about straights which don't include either of your hole cards - do they count? And again, do you mean 'a straight or better' or 'a straight but nothing more'? If the latter, what about a straight flush?
The problem with many of these "What are the chances of that?" questions is that they are not strictly defined, so it's hard to give a proper answer. But usually, if you can define the question sufficiently clearly, the approach is either to use probability (as with the 'set' example above) or brute force with the help of a computer.
To be specific, If you have A3 what is the probablity of a 2, 4, and 5 running out? Apr 16 at 7:01
Also why do you not take into account the cards that are dealt out when you calculcate the probability? 48/50 when the cards left in the deck are going to be less than 50. Apr 16 at 7:06
Re your second question: the cards dealt to other players don't affect the calculation because they are unknown. There are 50 unknown cards, of which two are your "set-makers". Any of the 50 cards is equally likely to be one of those two set-makers. Put another way, the first community (board) card to be dealt is equally likely to be any of the 50 remaining (unseen) cards. Of course, if any of your opponents show their cards, or muck face up, then that would change the probability of hitting a set for that specific situation, because you can then remove the visible cards from the calculation.– Neil TApr 16 at 9:31
Regarding the A3 question: this is a lot more complicated. If you would like a mathematical answer I think you should ask a new (precisely worded) question - adding the working to my answer above would make it too long, and diverge too much from the original question. If you just want the answer, I make it 2.69% that A3 will make a "wheel" straight by the river (including e.g. hands which also contain a flush or straight flush as well as "straights on the board").– Neil TApr 16 at 11:13