Technical support for Three Kingdoms (including mathematical modeling)

Three Kingdoms card categories,
There are a total of 108 three States, of which 30 are (21 are black, 9 are red, 70% are black), 15 are flash (all red), and 8 are peach (all red) 6 rivers and bridges, 5 good luck, 4 out of nothing, 3 South man invasion, 3 duel, 3 happy, 3 impeccable, 2 lightning, 2 grain fengdeng, 2 kill with a knife, 10 thousand Jian Qifa, 1 Taoyuan; A total of 20 pieces of equipment (60% black), of which + 1/-1 horse, 3 horses, 11 weapons (Zhuge crossbows 2, other 1) and 3 defense devices (2 octal Dipper and 1 renwang shield)

Probability determination of some basic conditions:

1. Which of the following is better? This problem is caused by human forms. Under normal circumstances, there are 70% million black kill, and the probability of gossip determination is 50% (54 red and black cards each), so the defensive strength of renwang shield is a little higher than that of gossip shield, but there are several special cases.

A) if you choose guo jia, you should choose gossip shield. All the cards you decide are yours. That is to say, when you have gossip shield, others give you a card. At this time, there will be a very interesting situation. Your teammates are equipped with crossbows, and then let them attack you. When you decide to fail and get the blood to touch your cards, you can kill and give it to your teammates, A little abnormal ......

B) If Sima Yi is present, you should consider whether he is a teammate with you. If yes, you can consider gossip. If not, gossip will become garbage.

C) If Guan Yu is present and is not a teammate, pay attention to it. Guan Yu has 54 red cards to kill and 21 black cards to kill. For this character, he has 72% Red kill. You should switch your renwang shield to gossip.

D. If Zhao Yun is present and is not a teammate, pay attention to it. Zhao Yun has 45 kill (24 Red kill, 15 of which flash red). The probability of a Red kill is 53.3%, which makes renwang shield not easy to use.

E) The last thing to note is that there is no chance for Ren Wang shield to kill you. When Guan Yu and Zhao Yun kill you, they will basically kill you. So from a practical point of view, we met these two people, renwang shield is a decoration.

2. What is the probability of being split by lightning? Haha, when Sima Yi is present, it is equivalent to a lightning rod on your head, or it is equivalent to a lightning rod on your head. there is basically no probability. Under normal circumstances, a total of 17 cards (five of them are killed, four weapons, one defensive device, and one horse + one horse) are played between taotao 2 and taotao 9, 6 Tips), the probability is 15.74%. From the mathematical expectation point of view, that is, touch 6 cards, one of which is, in other words, basically, there will be a lightning rod in the six rounds of determination. In actual games, the probability of weapons, armor, and horses being equipped is reduced to 10.2% even if they do not appear in the pile of cards.

3. What is the probability of being unhappy? There are 27 red peaches in the 108 cards, one of which is a weapon, one plus one horse, and one-one horse is a red peach. The probability of escaping from happiness is 25%, however, in actual games, weapons and horses will be equipped, so the chances of escaping will be reduced to 22.2%.

4. What cards should I leave when there is a flash in my hand? This is also determined by the field of view,

A) If you have security equipment, you can choose to kill it ...... ^-^, Haha, many people will forget their equipment.

B) if it is Ji, select "black" to kill. "Black" can be killed or flashed. Similarly, if it is Guan Yu, the flash will be left, and all the flash will be red, which can be used as a kill.

C) For other people, consider the location on the field and your role. If you are the primary Company, you will be beaten. If your side is always an enemy, keep it in the dark. In addition to these two cases, we also need to consider whether there is Cao on the court. This kid is so horrible that three nanman.com intrusions and one wanjian Qifa, he will spread into six nanman intrusions and two arrows, and the probability of nanman infiltration is 75%. If the attack is not impeccable, consider killing it to avoid being killed by Aoe.

Characteristic Analysis of various characters,
Explosive growth role,

1. Huang yueying, an amazing player. Before discussing the explosive power, I would like to make a statement here. The explosive power here is the explosive power in the beginning. I personally feel that the explosive power in the later stage is of little significance and the situation is complicated. I can't draw a conclusion. Huang yueying's explosive power is the biggest personal experience. He still uses numbers to talk about it. There are a total of 35 tips, 3 of which may not be usable (1 Taoyuan, 2 with a knife ), all of them can be used here. Among the 35 tips, there are four out of nothing (you can change to 3), five good luck, two five grain fengdeng (you can change to 2 ), you can change one other tip. For Huang yueying, one tip card is equivalent to 4/35*3 + 7/35*2 + 24/35*1 = 1.43 cards. When Huang yueying arrives at the rear of the game, there will be six cards in the hand, which are arranged and combined for calculation. The following conclusions are obtained,
A) the probability of 0 Tips is 8.9%.
B. The probability of one tip is 27.5%.
C. The probability of two tips is 33.8%.
D. The probability of the three tips is 21.3%.
E) the probability of four tips is 7.2%.
F) the probability of five tips is 1.2%.
G) the probability of 6 tips is 0.85%.
Here, we can get the expected mathematical value from the increase in the actual number of yellow moon English cards,
1*0.275 + 2*0.338 + 3*0.213 + 4*0.072 + 5*0.012 + 6*0.0085 = 1.913
Considering that each tip is equivalent to 1.43 cards, Huang yueying's number of cards should be 1.913*1.43 = 2.74, nearly three cards. Here we can define Huang yueying's explosive power as 2.74 +. Here, the ++ contains two meanings,

1) the maximum value of an instant explosion will be very large. If you see almost 20 cards in Huang yueying's hand, don't be shocked. That is completely possible.

2) the damages caused by the use of these brands cannot be ignored. After Huang yueying only uses all the tips, it is often seen that the injured primary company has only a few cards and is happy ......

Additionally, I came across many masters and equipped them with weapons to cut down the yellow moon. I want to consume the yellow moon's hand card or give at least a warning, this is actually a wrong approach. Do not equip yourself with weapons at least in the first round, and do not fight anyone at will. When it is the turn of soy sauce and purple, you can suppress the chance of the outbreak of porn and moon by suppressing the knife killing and peach garden jianyi tips.

2. Ji, the best player. This is easy to calculate. There are half of the 108 cards in red and black, that is to say, the probability of each successful decision is 50%. Here we also make a mathematical expectation,
1*0.5 + 1*0.52 + 1*0.53 + 1*0.54 + 1*0.55... + ......, We can get like 1. Similarly, we define the explosive power of Ji as 1 +. Here, + implies that there may be a lot of RPs in hand, it may also be judged by Sima Yi to get more.
PS: The 81-floor crosis supplement: If the card model is not infinite, but calculated by 108 pieces, the expected value is 0.98112 ~

3. Huang Gai, an outstanding player in the early stage. This makes it easier to calculate. You can use four cards, touch two cards, and touch six cards with self-loss blood at three o'clock. A total of 12 cards are available, with a low chance of having none, this is ignored. The probability of taotao burst is 7.4%. If taotao burst, it is equivalent to one more card. Therefore, the mathematical expectation of taotao burst is 2*0.926 + 3*0.074 = 2.074, with a total of 3 points of blood, therefore, the total number is 6.22. Because the yellow cover card is basically fixed, and there are only 12 cards in total, there is basically no more, and the 12 cards can do more limited, and in case they are not made, the next round will basically disappear. According to the principle of changing 1 blood to 2 cards, we can define that the explosive power of the yellow cover is 0.22, but because of the relatively stable explosive power, we can use 12 cards to blow the crossbows, therefore, it is also a favorite role of players. Here, by the way, the LORD met Huang Gai's strategy. Of course, it is the opposite to dealing with Huang yueying. We should try to deduct the blood volume of Huang Gai to make it lose its ability to break out.

4. Lu Xun, later-stage explosive player. Lu Xun is a later burst-style player. Such figures are suitable for being an insider. The company's skills allow him to stand alone in the future. The probability of a company's outbreak is the highest for all tips, kill, and peach. Here we will briefly describe the maximum and minimum probability definitions,

1) maximum chance: Kill (with crossbows), peach (with blood loss on your own), hand-picked sheep, Peach Garden righteousness, lightning (will hurt yourself), killing with a knife, duel (will hurt yourself), nanman.com intrusion (hurt teammates), Wan Jianqi (hurt teammates), other tips and equipment, only Flash and impeccable cannot be used, outbreak probability is 83.3%

2) minimum probability: 37 tips and equipment can be used with a probability of 34.4%
Obviously, the probability of Luxun's later outbreak is unstable, depending on the form on the field. Because he is a later player with unstable outbreak probability, we will not compare the explosive strength indicators here.

Wuzhi

5. Guan Yu, a killer player. Of the 108 cards, half of them are red and 21 are black. The total number of cards can reach 75. It can be said that they will not be killed.

6. Zhao Yun, both attack and defense players. 45 pieces of kill/flash, basically everything is competent.

7. Zhang Fei, who is a lethal and terrorist player, has eight spears and is basically invincible.

8. lub, the injury contestant, basically fell short of blood.

Other articles
9. daqiao, Cap contestant. There are 27 blocks. The result of the arrangement and combination shows that there is only one block with a probability of 17%. The probability of 83% can be at least one person. Cooperating with Sima Yi is a nightmare for the opponent ......

10. Hua Yu, a supporter. This character will not be analyzed. If you don't play the cards, 50% is peach. When you play the cards, 100% is peach, which is simply a cowhide.

11. Gan Ning: Card breaker. 54 in black colors are used as River bridges, and one Red River Bridge is used. Basically, 50% of the River bridges are difficult to split.

There are basically no probability issues for other figures, so we will not discuss them here.
Very powerful ~~~
More powerful ~~
Abstract:
This article calculates some interesting probability results in the Three Kingdoms Kill, including: the expectation of Edison Luo is 1, which is equal to the one confirmed by Zhou Yu. If only the two skills of Luo Shen and Ying Zi are compared, the former is dominant. Huang yueying's expectation is not as high as he imagined, but only 0.798. Huang Gai's initial average hand number is about 15, and Zhou Tai's most likely death time is 5th. The average calculated result shows that Zhou Tai's average death time is 5.43.

Introduction:
In today's probability theory class, when I talked about geometric distribution, I couldn't find a good example and stood on the podium, But I was immersed in a wonderful memory. It was a dark night at the foot of the lingshan mountains. I acted as the master of Ji, And I was defeated in the conspiracy. When the master of the blood looked at the Red Blood and smiled, I started Luo Shen, the following sequence appeared on the board: black, black ...... The Lord's father-in-law shouted at the end with a sigh of hate. With admiration for Ji, I tried to calculate the expectations of JI Luo's God and found that the results were different from what I expected. So I analyzed other RP-type heroes, then we have this hydrology.

1. Comparison Between Yan Ji and Zhou Yu

Yan Ji bo xing's beauty
Luo Shen: When the round starts, you can take the initiative to determine. If the result is a black color, it is deemed that the decision is successful, the decision card will be closed to the hand card, and you can continue to determine; if the result is a red color, it is regarded as a failure and the skill is terminated. it can be determined continuously until it is unwilling to continue or the result turns red.

Governor Zhou Yu
Yingzi: Touch three cards in the touch phase

Gong Jin's skills are clear at a glance. He needs to touch more cards, and Ji will be more arrogant. In summary, in addition to the two cards, he can still touch them until he finds them red. We have seen the ghost that has been suffocated by red cards, and the ghost that has been black cards like rain. To figure out the value of Ji's skills, let's calculate Ji's expectation for the number of additional cards.
According to the knowledge of probability theory, Ji's card number X is a discrete random variable that obeys the geometric distribution. The parameter P is 1/2. If a is used to indicate the expected number of additional cards

The summation of this series is very simple.

This shows that Ji's expectation is 1, which is equal to the expectation of Zhou Yu (I think Ji's expectation is smaller). From the expectation alone, there is no difference between the two. However, considering the unique nature of the Three Kingdoms Kill, especially the existence of ZHUGE crossbows and other devices, when there are many cards, the power increases very fast, and the players have a psychological pursuit of risk.

Nature 1. In general, the function U of the hand card's utility function about the number of hand cards X is a monotonically increasing convex function.

Using the convex function, we can easily prove that the hand effect caused by Ji's skill is greater than that of Zhou Yu. That is to say, if only Luo Shen and Ying Zi are compared, the former is dominant.

2. More hate for Yue Yingjie

Huang yueying's retired chivalrous lady
Ji Zhi: When she uses a card, she immediately feels a card from the top of the card and adds it to her hand.

I believe that everyone has seen the experience of seeing Huang yueying fall asleep on a single machine. With the above results, we can also try to calculate Huang yueying's expectations. As nothing in the tip is very appealing, we will discuss this issue separately. First, we make the following assumptions to simplify the calculation:

Suppose 1. The Lottery model is not put back. That is to say, at any time, from the top of the card stack to the possibility of a card * p * is the proportion of the number of cards in the total number of cards. This is not only for your own experience, but also ignores the possibility that someone else has the cards you want. We have used this assumption in Ji's discussion. This assumption leads to a very small error. In fact, the expectation for calculation is generally very small, around 1, so the probability of finding many cards is very small, which is consistent with our assumptions.

First, we will discuss the impact of inanimate players on average players.
P * indicates the probability of finding a card that has nothing to do. p * = A */S, A * indicates the number of blank pictures, and s indicates the total number of pictures, the probability of at least one of the two cards that have been discovered by an employee is 1-(1-p *) × (1-p *). Because p * is very small, it is equivalent to 2 p *. Actual situation a * = 4, S = 109, p * = 0.037. define n times of inactive cards as having n + 1 in the case of inactive cards, so from the above reasoning we get:

Nature 2. High-Order students can be ignored.

Next we will start to calculate the expected number of additional cards for Huang yueying. Here we also want to make a hypothesis that Huang yueying will use all the available tips, while ignoring the impact of all the five-grain fengdeng (Too collapsed ). First, calculate the number of additional pieces of B after using a tip. Use P to indicate the probability of finding a non-delayed tip in the next section of Huang yueying (excluding the possibility of being out of nothing ). Based on the properties similar to geometric distribution, we can obtain the relationship:
B = 1 + P x B + P x (3 + 3b)
Solve B's problem:
B = (1 + 3 P *)/(1-p-3p *)
In the game, P = 18/109 = 0.165. calculate B = 1.53 Based on the calculator. We can start to calculate a. (This is a comprehensive consideration of the general kit and a huge project out of nothing ...)

A = P x 2B + 2 p x B + P x 2 (2 + B) + 2 p x (2 + B) -2 p × P * (2 + 2B)
= 0.798

The calculation result is surprising. Huang yueda's expectation is not as high as imagined, only 0.798! In contrast, it is not as good as Zhou Yu and Mink Chan to determine one, and Ji's average one. For this question, we can explain it through nature 1. From B> 1, we can find that once Huang yueying finds a tip, his potential is very good, and Huang yueying is terrible in the first round of six cards. Huang yueying exchanged a lower average value for a higher risk, that is, explosive growth.

3. Gongjin, please spur me!

In the Three Kingdoms Kill game, there are always some evil people who love a violent hero.

Yellow cover light for the country
Bitter meat: You can take the initiative to reduce your physical strength and build two cards from the cards. You can use this skill multiple times in your own round.

With the previous analysis, the calculation of the yellow cover is much simpler. Make the following assumptions:
Suppose 2. (violent yellow cover hypothesis) the yellow cover will change itself to a drop of blood whenever possible.
So there are inferences.
Inference 1. All the peaches and Taoyuan Meanings of the violent yellow cover are equivalent to those of nothing.

The Probability p * = 13/109 = 0.119. ignoring the possibility of simultaneously finding three "out of nothing" and ignoring the high-order "out of nothing", the average number of hand cards that the violent yellow cover can obtain in the first round without being hit is approximately:

A = 4 + 2 + (4-1) × 2 + 12 p ** (1-p *) ^ 11 × 2 + (12 select 2) p * ^ 2 × (1-p *) ^ 10 × 4
= 12 + 1.766

During the calculation, it is found that due to the large initial number of sheets in the hands, p * is not very small, and the impact of the higher order "out of nothing" cannot be ignored, so a is only a lower bound of the yellow hand card, an estimation is provided based on various factors. The initial average hand number of the yellow cover is about 15! Imagine how much a man who sat down at the home of the master shouted "Heaven is dead, and heaven is standing up" as a threat to the master. (In the future, we should not despise the violent pornography ...)

4. Those who believe in spring have to live forever

No more.

The Battle of Zhou taili
Unyielding: at any time, when your physical strength is reduced to 0 or lower, each point of physical strength is deducted: a card is shown above the card stack and placed on your role card, if the number of points on this card is different from any other card on your role card, you will not die. At this time, the computer will be regarded as in the near-dead state.

The most disgusting little strong in history, I used to hold a crossbows in hand as a yellow cover to kill to the master, and was intercepted by the loyal minister Zhou Tai half-way, finally, it was chopped by one of them. From then on, how Long Zhou Tai can live has become a constant mystery in the rivers and lakes...
Compared with the previous two problems, Zhou Tai's calculation is not complicated and does not involve any lements. As long as the basic knowledge of probability theory is used. Use s to represent the total number of cards (S = 52 × 2 = 104 here). From the very moment of death, an indicates that after the n-th decision, the number of cards is still standing, DN indicates that Zhou Tai died in the N decision. We have:

The relationship between an and DN is as follows:

Enter the data to calculate the value list:

And images:

Conclusion: Zhou Tai is not that hard. The maximum likelihood of death for Zhou Tai is 5th judgments, and the average calculation result is that the average death time for Zhou Tai is 5.43 (I used to believe that this is 7 deaths ). Finally, we know how Long Zhou Tai can live.

Discussion:
This article provides several enlightening probability results for RP-type heroes. Among them, Ji and Zhou Tai's computation is basically accurate, and the discussions on nothing are similar, without considering the impact of grain fengdeng, there is still a great deal of uncertainty in Huang Gai's calculation. We hope that some cool people can give Huang Gai more reasonable computing results.

References:
[1] Introduction to probability theory, Wang renguan, Peking University Press, April 1994, 1st
[2] Three Kingdoms promotion brochure, Yoka Games