Chapter 9: The Three Door Problem

Li Renshu thought for a moment, then said, "You mean to say that you can see through this linguistic trap the designer buried in the rules, and then fire all six shots at yourself?"

What everyone else saw was the complete set of 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, and the five empty chambers are distributed in random positions within the cylinder.]

This clause was a pure linguistic trap.

Anyone who knew this could indeed clear the game without injury.

Wang Yongxin immediately voiced his objection: "But that’s only because we have a god's-eye view.

"Assuming we woke up knowing nothing, and were told this rule while our lives were under threat, the vast majority of people would be unable to think as rationally as you describe.

"To think you could realize such a trap is to have a bit too much faith in your own rationality."

Cai Zhiyuan shook his head: "No, I believe that even without noticing this trap, it wouldn't hinder one from clearing the game.

"Let’s break down the probability involved here.

"First, the sum of the distances between the iron block mechanism and the sides of the player's head is 6cm, meaning an average of 3cm on each side. For every shot fired at the innocent person, the iron blocks on each side move inward by 1.29cm.

"This means the first two movements cause no injury, and the third movement causes damage far less severe than the subsequent ones.

"However, for the fourth, fifth, and sixth movements, each one causes increasingly severe damage to the head, with the danger level increasing exponentially; by the fifth movement, death is almost certain.

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

"Assuming the mechanism kills you after five movements, we can roughly treat each movement as accumulating 1/5 of a 'death progress bar.' Of course, the mortality rate of the mechanism isn't evenly distributed; it gets higher the further it goes.

"Firing a shot at yourself carries a 1/6 probability of death—that is beyond doubt. Firing at the other person has a 5/6 probability of being a blank, but the mechanism's movement still adds 1/5 to the death progress bar.

"The actual risk of firing at yourself versus firing at the other person is roughly the same.

"Since you could be crushed to death after five movements, we must choose to fire at least two shots at ourselves.

"Assuming there really is one live round in the gun, the probability of any given shot being live is 1/6; which two shots you choose to fire at the other person doesn't actually affect the game's outcome.

"But psychologically, it’s definitely best to choose the first two shots.

"Because in practice, if the previous shot wasn't a hit, the probability of the subsequent shot increases accordingly, which creates immense psychological pressure.

"For example, if the first shot is a blank without you knowing it, the probability for every subsequent shot becomes 1/5. If the second shot is also a blank, every subsequent shot becomes 1/4, and so on.

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

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

"This hint is simply too merciful—isn't this just the classic Monty Hall problem?

"This means the probability of the fourth shot containing a live round is still 1/3, while the probability of the sixth shot containing a live round becomes 2/3. If you can make a rational decision, you should still fire the fourth shot at yourself.

"If you can grasp this, then even if you choose to fire at the innocent person for the final shot, combined with the one shot fired at the innocent person in the first three, the iron block mechanism will move at most twice.

"In terms of distance, you wouldn't even get a scratch.

"Not to mention, from our god's-eye view, we know there isn't a single bullet in the revolver, so the possibility of being killed by a bullet doesn't exist at all."

The group fell into a temporary silence.

Jiang He, a newspaper editor, frowned and asked: "I mostly understood the first part, but I didn't get the last three shots. What is the Monty Hall problem?"

Cai Zhiyuan was surprised: "You don't even know that?

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

"It originated from a foreign television show

"There are three closed doors in front of a contestant. Behind one door is a car, which you win if you pick it; behind the other two doors, there is nothing.

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

"At this point, the host opens one of the other two doors, revealing it to be empty—no car. Note that the host doesn't open a door at random; because he is the host, he knows from the start which door has the car behind it. The door he opens is one he already knew was empty.

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

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

Jiang He thought for a moment and said firmly: "I wouldn't. I trust my first instinct.

"Besides, isn't the probability of the car being behind any given door 1/3? What difference does it make if I switch or not?"

Cai Zhiyuan shook his head: "That’s where you’re wrong.

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

Jiang He was stunned: "Huh? Why?"

Cai Zhiyuan explained: "That’s why the Monty Hall problem became a classic probability puzzle; it seems simple, but it is highly counterintuitive.

"It’s normal for you to be confused, because at the time, this problem sparked fierce debate, and many scientists and scholars even opposed the conclusion.

"Proving this is quite complex, but I have a way to explain it that’s easier to understand

"Suppose we increase the number of doors to 10,000. One door has a car behind it, and the other 9,999 doors have nothing.

"You pick one door, and the host, who knows the car's location in advance, opens 9,998 of the other empty doors, leaving only one left.

"Now the host asks you: Do you want to switch doors?

"Would you switch this time?"

Jiang He thought for a moment: "I would."

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

Jiang He pondered: "With ten thousand doors, it’s almost impossible to pick the car correctly on the first try; the probability is one in ten thousand.

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

"So the car must be behind the other door."

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

"No matter how the host opens the doors, the original door's probability remains unchanged because it was chosen at the start, but the probability of the other doors increases.

"So, let’s go back to the original Monty Hall problem: the probability that the door chosen by the contestant has the car is 1/3. We treat the other two doors as a single unit; the probability that the car is behind one of them is 2/3.

"After the host eliminates one door, the probability of the two-door unit having the car becomes equivalent to the probability of the remaining door having the car.

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

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

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

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

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

"You have to choose the shot with the lower probability to fire at yourself, and the one with the higher probability to fire at the innocent person."

Cai Zhiyuan praised: "Exactly. You’re very smart; that’s exactly how it is."

The group fell into a brief silence, all digesting what Cai Zhiyuan had just said.

After thinking it over carefully, Fu Chen said: "So, according to this analysis, 'Redemption Roulette' is actually a game that tests 'linguistic sensitivity' and 'probability'?

"But can you really get an S-rank just because of that?"

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 as a matter of course: "I could."

Li Renshu agreed: "I think so too, and it probably wouldn't be because of any probability problems."

Cao Haichuan seemed to have a nicotine craving; he subconsciously reached for a cigarette but held back in the end.

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

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

(End of chapter)