Three Doors Problem

Li Renshu thought seriously for a moment, then said, "You mean you could see through the language trap buried in the rules by the designer, and then fire all six shots at yourself?"

What everyone had seen was the complete set of rules. In other words, they knew that the live round was in the innocent person's pocket, not in the revolver's cylinder.

The rule stated: [The revolver's cylinder contains six chambers, five of which are empty and distributed randomly throughout the cylinder.]

This was purely a language trap.

Anyone who realized that could indeed clear the game without injury.

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

"Suppose we woke up knowing nothing and were told this rule while our lives were under threat. The vast majority of people wouldn't be able to think as rationally as you described.

"You think you'd notice that kind of trap? That's putting far too much faith in your own rationality."

Cai Zhiyuan shook his head. "No. I think that even if one failed to notice the trap, it wouldn't prevent them from clearing the game.

"Let's sort through the probabilities here.

"First, the total distance between the Iron Block Mechanisms on either side of the player's head is six centimeters, which means an average of three centimeters per side. Every time the innocent person is shot, the iron blocks on each side move inward by 1.29 centimeters.

"That means the first two movements cause no harm, and even the damage caused by the third movement is far less severe than that of the later ones.

"But the fourth, fifth, and sixth movements would each inflict increasingly severe damage to the head. The danger rises exponentially. By around the fifth movement, death would be virtually certain.

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

"Suppose five movements of the mechanism would kill someone. Then we could roughly treat each movement as filling one-fifth of a death progress bar. Of course, the mechanism's fatality rate isn't evenly distributed; it gets higher the later it goes.

"If you shoot yourself, there's a one-in-six chance of dying. There's no question about that. But if you shoot the other person, there's a five-in-six chance of firing a blank, and the mechanism moving would likewise add one-fifth to the death progress bar.

"The actual risks of shooting yourself or shooting the other person are roughly the same.

"Since being crushed to death becomes possible after five times, we must choose at least two shots to fire at ourselves.

"Assuming there really is one live round in the gun, every shot has a one-in-six chance of being the live one. Which two shots we choose to fire at the other person doesn't actually affect the outcome of the game.

"But psychologically, the first two shots are definitely the best choices.

"Because in actual practice, if the previous shot didn't hit, the odds for the next shot would rise accordingly, creating enormous psychological pressure.

"For example, if the first shot fired a blank without your knowing it, the chance for every subsequent shot would become one in five. If the second shot was another blank, the chance for each remaining shot would become one in four, and so on.

"So whether you think the probability of each shot is the same or different, the first two shots should be prioritized for shooting yourself.

"When it came to the fourth shot, there would be a new hint: the fifth shot was a blank.

"That hint was simply too merciful. Wasn't this the classic Three Doors Problem?

"That meant the fourth shot still had a one-in-three chance of being live, while the sixth shot's chance of being live became two-thirds. If one could make a rational decision, the fourth shot should still be fired at oneself.

"If one understood that, then even if the final shot was fired at the innocent person, together with the one shot among the first three that was fired at the innocent person, the Iron Block Mechanism would move at most twice.

"Judging by the distance, there wouldn't even be a superficial wound.

"Besides, from our omniscient perspective, we know there isn't a single bullet in the revolver at all. So the possibility of being shot dead simply doesn't exist."

Everyone fell silent for the moment.

Jiang He, the newspaper editor, frowned deeply and asked, "I more or less understand the earlier part, but I don't understand the last three shots. What is the Three Doors Problem?"

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

"All right, then I'll explain it simply. It's actually a very classic probability problem.

"It originated from a foreign television program:

"There are three closed doors before a contestant. Behind one is a car, which they can win if they choose it. Behind the other two is nothing.

"The contestant picks one of the doors but doesn't open it right away.

"At that point, the host opens one of the other two doors. There's nothing behind it—no car. Note that the host does not open a door at random. He is the host, so he knows from the start which door has the car behind it. 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 about it, then said with certainty, "No. I trust my first instinct.

"Besides, isn't the chance of a car being behind each door one-third? What difference would switching make?"

Cai Zhiyuan shook his head. "Then you'd be wrong.

"Because the probability that the original door has the car remains unchanged at one-third, but the probability that the other door has the car becomes two-thirds. You should switch."

Jiang He froze. "Huh? Why?"

Cai Zhiyuan explained, "That's why the Three Doors Problem became a classic probability problem. It looks simple, but it goes strongly against intuition.

"It's normal for you to be confused. Back then, this question sparked intense debate. Even many scientists and scholars opposed this conclusion.

"The proof for this problem is rather complicated, but I have an easier way to understand it:

"Suppose we increase the number of doors to ten thousand. One door has a car behind it, while the other 9,999 have nothing behind them.

"You choose one door. The host knows where the car is in advance, then opens 9,998 of the other empty doors, leaving only one other door.

"The host asks you again: Do you want to switch doors?

"Would you switch this time?"

After a little thought, Jiang He said, "Yes."

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

Jiang He lowered her head in thought. "Out of ten thousand doors, accurately choosing the car right away is practically impossible. 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 the other door."

Cai Zhiyuan nodded. "Exactly. Once the number of doors increases, the problem becomes easy to understand.

"No matter how the host opens the doors, the probability of the original door remains unchanged because it was selected from the beginning, while the probabilities of the other doors rise.

"So let's return to the original Three Doors Problem. The contestant's chosen door has a one-third chance of containing the car. We treat the other two doors as a whole, and their probability of containing the car is two-thirds.

"After the host eliminates one door, the probability that the two-door whole contains the car becomes equivalent to the probability that the remaining door contains the car.

"That door's probability changes from one-third to two-thirds."

Fu Chen understood. He nodded slightly, looking thoughtful.

"So when this game reached the last three shots, the updated rules on the television essentially turned it into the 'Three Doors Problem.'

"The upcoming fourth shot was the originally chosen door; the fifth shot was the door eliminated by the host; and the sixth shot was the remaining door.

"The host asking whether to switch doors was equivalent to the player deciding whether to switch the fourth shot for the sixth shot.

"Of those two shots, they had to choose the one with the lower probability to fire at themselves, and the one with the higher probability to fire at the innocent person."

Cai Zhiyuan said approvingly, "That's right. You're very smart. That's exactly it."

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

After thinking it over seriously, Fu Chen said, "Then based on this analysis, Redemption Roulette is really a game that tests 'sensitivity to wording' and 'probability'?

"But could that alone earn 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 it?"

Cao Haichuan nodded as if it were only natural. "Yes."

Li Renshu agreed. "That's what I think too, and it probably wouldn't be because of any probability problem."

Cao Haichuan seemed to be hit by a cigarette craving. He instinctively reached for cigarettes, but ultimately held himself back.

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

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