The Three Doors Problem

Li Renshu pondered for a moment before saying, "You mean you can see through this linguistic trap laid by the designers in the rules, and then shoot yourself six times?"

The others saw the complete rules, meaning they knew the live round was in the innocent person's pocket, not in the revolver's cylinder.

The rule stated: [The revolver's cylinder has 6 chambers, with five empty chambers distributed randomly.]

This was purely a linguistic trap.

Anyone who knew this could indeed pass the game without harm.

Wang Yongxin immediately objected, "But that's because we have an omniscient perspective.

"Suppose we woke up with no knowledge, were told this rule under threat to our lives, and the vast majority of people wouldn't be able to think rationally as you say.

"To think you can recognize this trap is to have too much faith in your own rationality."

Cai Zhiyuan shook his head, "No, I don't think failing to notice this trap prevents us from passing the game.

"Let's go over the probability issues here.

"First, the sum of the distances between the Tekkai mechanism and both sides of the player's head is 6cm, meaning an average of 3cm on each side. Shooting the innocent person once causes the Tekkai on each side to move inward by 1.29cm.

"This means the first two movements are harmless, and the damage from the third movement is far less than from subsequent ones.

"As for the fourth, fifth, and sixth movements, each will cause more severe damage to the head, with the danger increasing exponentially. It's almost certain death by the fifth movement.

"So, when considering the risk of death, we must consider not only 'being hit by a bullet' but also 'being crushed by the mechanism.'

"Assuming the mechanism causes death after five movements, we can roughly consider each mechanism movement as accumulating 1/5 of the death progress bar. Of course, the death rate from the mechanism isn't evenly distributed; it's higher in later stages.

"Shooting yourself has a 1/6 chance of death, no doubt about it. Shooting the other person has a 5/6 chance of being a blank, and the mechanism movement also contributes 1/5 to the death progress bar.

"Shooting yourself or the other person carries roughly the same risk.

"Since death is possible after five movements, we must choose at least two shots to fire at ourselves.

"Assuming there is indeed a live round in the gun, the probability of being shot each time is 1/6. Choosing which two shots to fire at the other person doesn't affect the game's outcome.

"However, psychologically, it's definitely best to choose the first two shots.

"Because in practice, if the first shot misses, the probability of subsequent shots increases, which brings immense psychological pressure.

"For example, if the first shot is a blank without your knowledge, then each subsequent shot's probability becomes 1/5. If the second shot is also a blank, each subsequent shot becomes 1/4, and so on.

"So, regardless of whether you believe the probability of each shot is the same or different, the first two shots should prioritize firing at yourself.

"By the fourth shot, there will be a new prompt: the fifth shot is a blank.

"This prompt is too merciful. Isn't this the classic Three Doors Problem?

"This means the probability of the fourth shot containing a live round is still 1/3, while the probability for the sixth shot becomes 2/3. If one can make a rational decision, the fourth shot should still be fired at oneself.

"If one understands this, then even if the last shot is chosen to be fired at the innocent person, combined with the one shot fired at the innocent person in the first three, the Tekkai mechanism will move at most twice.

"In terms of distance, it won't even cause a superficial wound.

"Moreover, from our omniscient perspective, we know there are no bullets in the revolver at all, so the possibility of dying from a bullet is nonexistent."

The crowd fell silent for a moment.

Jiang He, the newspaper editor, frowned and asked, "I understood the first part, but I didn't get the last three shots. What is the Three Doors Problem?"

Cai Zhiyuan looked somewhat surprised, "You don't even know that?

"Alright, let me explain it simply. It's a very classic probability problem.

"It originated from a foreign TV show:

"A contestant is faced with three closed doors. Behind one door is a car, which they can win if they choose it. Behind the other two doors is nothing.

"The contestant chooses one door but doesn't open it immediately.

"At this point, the host opens one of the other two doors, revealing nothing behind it. Please note, the host doesn't open a door randomly. Because he is the host, he knows from the beginning which door has the car. The door he opens is one he already knows is empty.

"The host then asks the contestant: Do you want to switch doors?

"If you were the contestant, would you switch?"

Jiang He thought for a moment and said confidently, "I wouldn't switch. I trust my first instinct.

"Besides, isn't the probability of a car behind each door 1/3? Does it make a difference whether I switch or not?"

Cai Zhiyuan shook his head, "Then you are wrong.

"Because the probability of the original door having the car remains unchanged at 1/3, but the probability of the other door having the car becomes 2/3. You should switch.

"Jiang He was stunned, "Huh? Why?"

Cai Zhiyuan explained, "That's why the Three Doors Problem became a classic probability problem. It seems simple but is very counter-intuitive.

"It's normal for you to be confused, as this problem sparked intense debate at the time, with many scientists and scholars even disagreeing with the conclusion.

"The proof is a bit complex, but I have a simpler way to understand it:

"Suppose we increase the number of doors to 10,000, with one door having a car behind it and the other 9,999 doors having nothing.

"You choose one door. The host, knowing the car's location, opens 9,998 other empty doors, leaving only one remaining.

"At this point, the host asks you again: Do you want to switch doors?"

"Do you want to switch this time?"

Jiang He thought for a moment. "Yes."

Cai Zhiyuan asked, "Then why did you decide to switch this time?"

Jiang He lowered her head and thought, "With ten thousand doors, it's almost impossible to pick the car right away. The probability is one in ten thousand.

"The door I originally chose definitely doesn't have the car.

"So the car can only be behind one of the other doors."

Cai Zhiyuan nodded. "That's right. Once the number of doors increases, this problem becomes much easier to understand.

"No matter which door the host opens, the probability of the originally chosen door remains unchanged because it was selected at the beginning. However, the probability of the other doors increases.

"So we return to the original Monty Hall problem: the probability of the contestant's chosen door having the car is 1/3. Let's consider the other two doors as a single unit; the probability of the car being behind them is 2/3.

"After the host eliminates one door, the probability of the car being behind the combined unit of two doors becomes equivalent to the probability of the car being behind the remaining single door.

"The probability of that door changes from 1/3 to 2/3."

Fu Chen understood. He nodded slightly, lost in thought.

"So, when the game reached the last three shots, the rules were updated on television, and it essentially became the 'Monty Hall problem.'

"The fourth shot, which is about to be taken, corresponds to the originally chosen door; the fifth shot is the door eliminated by the host, and the sixth shot is the remaining door.

"When the host asks if you want to switch doors, it's equivalent to the player deciding whether to switch their fourth shot for the sixth shot.

"You have to choose between these two shots, picking the one with the lower probability for yourself and the one with the higher probability for an innocent person."

Cai Zhiyuan praised, "Exactly, you're very smart. That's precisely how it is."

Everyone fell into a brief silence, digesting what Cai Zhiyuan had just said.

After thinking carefully, Fu Chen said, "Then, according to this analysis, the 'Redemption Roulette' is actually a game that tests 'text sensitivity' and 'probability'?

"But is that enough to get an S rating?"

Li Renshu seemed to realize something. She looked at Cao Haichuan.

"Officer Cao, if you were a player in this game, do you think you could survive?"

Cao Haichuan nodded matter-of-factly. "Yes."

Li Renshu agreed. "I think so too, and it probably has nothing to do with probability."

Cao Haichuan seemed to have a craving for a cigarette and instinctively reached for one, but ultimately restrained himself.

"Yes, I thought about it. There's no special reason why I could survive, as I don't understand probability.

"I simply couldn't bring myself to point the gun at an innocent person."