The top-notch movie queen wears books, disguises herself as a man and is handsome all over the world

Qin Jingyu saw Chen He taking out his extracurricular books and not answering the questions like the others. She glanced at the burrowed figure behind Chen He, and then she had a plan in mind. Her immature idea required a well-matched opponent.

The topic was casually changed, "Aren't you here to brush up on questions?"

Chen He asked back: "Aren't you also not using it?"

Qin Jingyu raised his eyebrows: "So, are you free?"

Chen He: "Huh???"

"I invite you to play a game." Qin Jingyu has now learned a lot about Shen Tu's polite invitation.

"game?"

"The cat catches the mouse."

Chen He: "What?"

It was obviously a very simple literal meaning, but why was it that Qin Jingyu couldn't understand it when it came out of his mouth?

"Have you heard of random matrix theory and transfer matrix?"

Chen He didn't know what the other party was trying to say, so he frowned and said, "I've only heard a little bit about it."

And he remembered that there are several different definitions and types of random matrices.

In order to let the other party participate in the game more deeply, Qin Jingyu began to explain to him that the right random matrix is ​​a real square matrix, in which the sum of each row is 1, and the left random matrix is ​​a real square matrix, in which the sum of each column is 1. A random matrix is ​​a square matrix of non-negative real numbers, with each row and column summing to 1.

Chen He looked at the formula written by her and was confused at first, but as he was lured step by step, he gradually began to understand.

Therefore, in the same way as mentioned above, a random vector, also called a probability vector, can be defined as a vector whose elements are non-negative real numbers and whose sum is 1.

Qin Jingyu found that Chen He was very smart. He would quickly think about and transform what she said.

Chen He suddenly realized: "The transition matrix can be used to express probability or change ratio, and the result of matrix multiplication can be used to predict the probability of future events."

Qin Jingyu snapped his fingers, "I understand that right."

These knowledge points were all novel to him, and Chen He was now completely interested, "So how do you play the cat-and-mouse game?"

Qin Jingyu had told him so much knowledge, and he certainly wouldn't be so stupid that the other party just wanted to play a literal cat-and-mouse game with him.

So he saw Qin Jingyu take out a mobile phone and set a timer, and then drew five adjacent grids in a row on the draft paper, and then used x's and circles to replace the roles of cat and mouse respectively.

At zero time, there is a cat in the first grid and a mouse in the fifth grid. When the timer increases, the cat and the mouse will randomly jump to an adjacent grid...

Then by analogy, when x and the circle belong to the same grid, it means that the cat has eaten the mouse, and the game will end.

"Want to play?" Qin Jingyu asked Chen He after explaining the rules.

"Play."

In the first round, Qin Jingyu was the cat and Chen He was the mouse.

Five minutes later, Chen He made a miscalculation and Qin Jingyu ate him.

In the second game, add one grid.

Chen He is a cat and Qin Jingyu is a mouse.

After twenty minutes of fighting, he couldn't catch the mouse. In fact, based on the advantages of random variables and fewer grids, the cat had a chance to eat the mouse.

To be precise, it wasn't that he couldn't catch the mouse, but that he couldn't catch Qin Jingyu. Her calculations were always one step ahead of him.

The stalemate continued for another ten minutes, which was more difficult than playing Go, so he gave up: "I give up."

Adding one more square, the initial value was 7, the roles of the two were reversed. This time, Chen He was caught in less than ten minutes.

Lost three games in a row.

Qin Jingyu still asked, "Do you want to continue?"

Chen He shook his head, "I need to go back and study it."

He is no match for the opponent. He cannot keep up with the calculation ability of the first prize winner in the mathematics league.

She felt it was a pity, but at present, it seemed that Chen He did not have a thorough understanding.

Just when she was about to end the game.

"Can I have a round?"

When the two heard the sound, they looked over and saw that it was Chen Shaochuan.

Chen Shaochuan had been sitting in the seat behind the two of them doing his homework. Although Qin Jingyu and Chen He spoke in a very low voice, they hardly affected anyone.

But when he heard the rules of the cat-and-mouse game explained, he was attracted, so much so that when Chen He gave up, he felt itchy.

Being socially awkward, he asked if he could join the game for the first time.

Of course Chen He knew the master of the physics class and the champion of last year's National Olympics. He looked at Qin Jingyu and then at Chen Shaochuan, "You want to play?"

Chen Shaochuan ignored him and just looked at Qin Jingyu.

The corner of the latter's mouth curled up slightly, and she knew that Chen Shaochuan would not be uninterested. She had taught Chen He for a long time and finally attracted this great master.

Qin Jingyu said to Chen Shaochuan: "Of course."

So Chen Shaochuan sat in Chen He's seat, and the latter took the things away very knowingly, but of course he had to stay and watch the PK between the two physics class masters.

In the first game, the initial value is 10, Chen Shaochuan is the cat and Qin Jingyu is the mouse.

The two filled their positions very quickly. Before Chen He could calculate the positions in his mind, they had already moved two spaces.

In less than three minutes, Qin Jingyu was caught.

This is the fastest round.

Chen He also realized the gap between him and these two people.

When Chen Shaochuan saw that he had caught Qin Jingyu, he was not happy at all, and said coldly: "You don't need to give in to me, you can increase the initial value."

When Qin Jingyu heard this, a flash of surprise flashed in her eyes. She originally wanted to let Chen Shaochuan go through it and understand the game, but she didn't expect that the opponent was not weak and could see through her at a glance.

Started to get serious: "Okay."

Increase the initial value to 100 and change the roles. Qin Jingyu is the cat and Chen Shaochuan is the mouse.

Both of them started to get serious about it, and the time stretched out instantly.

Chen He watched from the side and finally understood why this grid game is called a cat-and-mouse game. The cat chases the mouse. As long as the mouse is smart enough, it will always escape to the next safe hole under the cat's claws at the critical moment of life and death.

Until the manuscript paper he brought was not enough, night fell, and the manuscript paper he brought was not enough. When the game was about to be declared no winner and it was a draw, Qin Jingyu caught the mouse in the last step.

Chen He groaned, feeling it was a pity.

Chen Shaochuan's eyes behind the black-framed lenses were stunned for a moment, and then he frowned dissatisfied. He actually lost!

His voice was hoarse: "One more game."

Qin Jingyu said: "No need for the time being."

Chen Shaochuan showed a rare expression, "Why?"

Qin Jingyu: "Because I have grasped the rules."

"In fact, no matter what the initial state value is, the cat will eventually catch the mouse with a probability of 1. Let's use 100 grids as an example, and the limit is the steady state π = (x...99.1) So we need to calculate the long-term average or expected value of the random variable X, for each state and time, survival or not can be regarded as a binary variable."

Chen Shaochuan: "x=1 represents the survival state and x=0 represents the termination state. The state of x=0 does not contribute to the long-term average."

Since each state occupies a time step, the expectation of the mouse's survival time is that the sum of the probabilities occupied in all survival states and time steps will be...