If you are interested in professional poker, you will eventually realize that statistics are an important part of the game. A professional player should know what chances he has to get a certain combination. Based on these statistics, players make lists of the best hands at the beginning of the game, the worst hands at the beginning of the game, and the chances of getting a certain combination with the starting cards.
For example, we will analyze a rather popular statistical question that is used by many players “What is the probability that a five-card poker hand contains exactly one ace?”
People familiar with statistics will easily understand this issue, but we will try to make it simple for people without a degree in mathematics. Without delving into the formulas, when playing with a standard deck of 52 cards, the number of possible combinations of five cards is 2.598.960, which in turn is a huge number and shows the impossibility of calculating exactly what you will get.
C(52,5)=52!/(5!-(52-5)!)
Next, to calculate what is the probability that a five-card poker hand contains exactly one ace we need to find out how many of the huge number of combinations have only one ace. Fortunately for this case, there is a formula, we must multiply the chance of getting an Ace by the chance of getting four non-Aces. The standard deck of cards contains four aces and 48 cards that are not Aces. If you are wondering how the calculation is done, you can ask about game theory or probability theory, these calculations are one of the first you will learn. So, the number of successful outcomes will be 778.320, which is large even against the previous value.
C4,1 * C48,4 = 4!/(1!-(4-1)!) * 48!/(4!-(48-4)!)
Finally, to find the probability that a five-card poker hand contains exactly one ace we should divide the number of combinations when we will get only one Ace in the hand of five cards on the number of possible combinations of five cards. 778.320 / 2.598.960, by simplifying this action we get the following value 3243 / 10829, which in turn gives approximately 0.2995.
P=Success outcomes /All outcomes
So, we found the answer to the question: what is the probability that a five-card poker hand contains exactly one ace, it is 29,95 %. That is, based on chance, we can say that the probability of getting an ace on the table is slightly less than one in three. However, in a normal game, you should not forget about the presence of opponents who also have a chance to get an ace in their hand, and adjusting the percentages according to your hand, because if you have an ace in your hand, then there will be three of them in the deck, which will change the numbers to a lower. By such calculations, the chances of getting the majority of significant combinations were determined.