I prоbably will not go into combinatorics much and describe the calculations in detail. Just tell you about the results.
In total, there are 7776 variants for dropped dice, each of which is equiprobable (unless of course you roll all five dice). Accordingly, the probability of falling out of each combination is as follows:
Poker 6/7776 = 0.077%
Kare 150/7776 = 1.93%
Full House 300/7776 = 3.86%
Straight 240/7776 = 3.086%
Troika 1200/7776 = 15.43%
Two pairs of 1800/7776 = 23.15%
The pair 3600/7776 = 46.3%
Nothing 480/7776 = 6.17%
As you can see, the probability of falling of pairs (one or two) or three is highest. In particular, the probability of getting "nothing" is much lower. Although it is much more interesting that the full house actually turns out a little more often than the straight, while it is valued higher.
However, all of the above is of little interest to the player - because if he had a certain combination, then the probability of this event is no longer important to him. Another thing is the probability of improving the available combination during the transfer.
Again, I will not consider all possible variants of transfer. And so it is intuitively clear that you need to transfer cubes not included in the available combination (unless there is no need to throw out poker or penalty). But here are some values of the probability of the desired result worth thinking about.
Kare - the probability to improve this combination before poker - 1/6 = 16.7%.
Full house - if you try to transfer a couple of it, then the probability of losing poker will be 1/36, and penalty 10/36. 11/36 = 30.5% is more than one fourth. I do not know about you, but if I had a penalty and the opponent had a full house with a triple of greater importance, I would have thought more than once before raising the bet. But if he has a couple in a full house of greater importance than I have in the penalty, then the probability of throwing "more" penalty (moving the other three), he will only have 5/72 - that is, to transfer in any case you need a pair with the calculation for poker .
Straight is actually a pretty fucking deal, since the probability of improvement is extremely low. More precisely, it is the same as if you were throwing cubes anew. The only advice, if you still hope to improve the straight - move four dice, leaving one on which the maximum value. The probability of dropping the best cоmbination from this will not change, but if it falls out, it is with this value.

On poker math, whole books are written, so you can not say that it is needed for one thing, because this is one of the main foundations of poker. But there are two main aspects
1. Competent calculation of bank's chances
Let's start immediately with the analysis of the situation:
When playing 1 on 1, the size of the pot is $ 100, your opponent is betting at $ 50. How often do we need to call so that the action is not a minus one?
In this situation, your inner clue can give the wrong guess, and say that our hand here needs to have 50% equity. But this will be far from the truth, in fact, in order to play such situations in plus, we will have enough hands and with 26% to win the hand (if the hand will have 25% equity - we will play such a situation at zero on the distance).
In fact, the formula here is quite simple: we divide the opponent's bet in the big blinds by the number of big blinds in the pot, after the opponent's bet (TP) + the opponent's betting rate (AC), and multiply by 100%
As applied to our example (the big blind is $ 10):
Percentage in bank odds
If you find a phrase like: we need here equity 3k1, then do not be scared. In fact, these are the same calculations, but presented in a different form. This designation will be more close to those who know firsthand about bookmaker odds. That is, it means that we need to win once in three, so that the call was considered a plus, that is, we need at least 33.3% equity, as we said, everything is extremely simple!
2. Properly calculated bluff
But these are just the basics!
For a deeper analysis of distribution, you will not need a hefty share of time and patience. For example, if you need to calculate the optimal range for a specific situation, or optimal sizing for a bet on a value, or a bluff, and much, much more. This topic is not even on one book, not that article. And even those who play sky-high limits, although they have a huge knowledge base about poker mathematics, will not be able to understand all its subtleties.
But the task of this article was not to delve into the theory of probability, or to understand the basics of constructing ranges, but to acquaint you with the basics of poker mathematics. With these simple calculations, using them in the analysis of hands, and then during the session, you can significantly improve their results if you have not used them before. The most important thing is not to be afraid that it will be too difficult for you, and not to build yourself unnecessary obstacles in advance. Just start small, and along with understanding the game, you will get confidence in yourself and the results of your work, in the form of banknotes!

Poker, unlike other intellectual games, seems very simple to understand, especially for those who do not have much experience. Of course, it is much easier to win a poker novice compared to chess, because in chess, the influence of such factor as luck is minimized. With very big luck, even the most complete amateur who has just learned the rules of poker can even win a major tournament. But if you look at the whole picture, then in order to achieve real success in poker, to become a real professional, you need to spend a lot of time on mathematical calculations. Far from all the regulars are enthusiastic about the idea, for hours they calculate the profitability of certain actions, although this gives a powerful impetus to development as a player.
But there are also players who have a different point of view on this account, that is, those who seriously claim that they do not use math at all in their game, and at the same time have positive schedules. This "intuitive" approach to the game and the truth is the place to be among poker regulars. So maybe it's true, why do you need to score your head with some formulas and calculations, when you can play yourself quietly "on the cheek"? Definitely, this kind of thinking will certainly be wrong in the future for a number of reasons:
First, players who do not use math in their game, but at the same time earn a stable income, are more likely to earn even more by using it.
Secondly, not at all the limits of the game will at least somehow a plus. At medium and high limits, you will often make mistakes, doing negative actions, well, and your opponents will gladly exploit them. Although at low limits there are some clever regulars that will collect value, using your faces in the game.
Third, even the most basic knowledge of poker mathematics will help you better understand the principles of the game, and develop the logic of decision-making.
Everyone who starts to apply mathematical calculations in the game, finds himself a few gross errors, the correction of which will save a lot of money over a long distance. To date, the surest way to find out how true this or that solution was is to check it with the help of calculations.

The phenomenal popularity of poker is based not only on its fascination but also on the possibility of getting a big win. This game gives you a possibility to become a rich and famous person!
Naturally, this development of the poker career scenario is quite rare, so poker pros who have earned millions by playing are almost equal to movie stars and celebrities in other areas of life. Their success, an interesting lifestyle and the ability to earn by doing their hobby motivates most new players.
Poker is not a game of luck, which has been proven more than once by the most successful poker players who can win even with the most trash starting hands. From this, it follows that only hard work, the desire to become a master of his game and full self-control at the poker table will help you to achieve success. There is no need to look for exact instructions on how to play poker, but there are tips on the sequence and set of actions because not all masters just play, many share their experiences, talk about their way to success. Each poker player has his own set of tricks, techniques, and secrets that make his style unique, but the fundamental basis for achieving a stable game always consists of the same components.
Regular practice
Each poker player understands that the study of theoretical materials will help to understand how to play poker, but you can really learn this knowledge only in practice.
Regular training is one of the main factors for improving skills since theoretical knowledge is only useful in their practical application, otherwise, they will remain useless in the player's head. However, it should be understood that not every game brings poker experience, new skills, and benefits.
Most experts are sure that gaming applications in social networks or games against the computer are a waste of time because the loss of training chips will not be a big upset, since it will not affect the well-being of a player in real life. So, you need to play using real money.

Psychology
Numerous books have been composed about poker psychology, so it is straightforward how critical this aspect is. Some authoritative poker players claim that the main thing in poker is psychology. It should be viewed from several angles:
An ability to influence yourself and to control your own emotional state.
An ability to notice certain signals in the opponent's behavior, including gestures and facial expressions.
An ability to influence an opponent to achieve your own goals.
One of the most popular books is called "The Psychology of Poker," which was written by Alan Schoonmaker. It is recommended to read by all experienced players and experts.
Additional information about opponents
Having information about what game style the opponent prefers, how often he watches the flop, what his usual range of hands and about many other nuances means to build the right tactics that can bring you a victory.
New players often have a very little understanding of how this information can be learned and how to apply it. But for a professional, it is necessary to analyze every situation in a game. Today there is no need to collect statistical data on your own. For this purpose, special programs of various complexity are intended. Not all poker rooms allow using them, but a separate software works separately from the client, so some information can always be obtained.
Poker Mathematics
All professional players base their gaming activities on clear mathematical calculations, so the expression that the basis of poker is mathematics should not be ignored. For new poker players, the mathematical aspect of poker seems to be the most difficult and frightening, with the result that they have to act intuitively. But everything does not work in this way.
There are many materials that help us to understand how to determine probabilities, outs and make other calculations.
Having spent some time studying the materials, the poker player will receive a much more successful game result.
So, do not be lazy and spend some time learning math processes in poker. You may become successful only by working hard.