Chapter 28: Probabilities of Hand Patterns

Wang Yongxin paused slightly before continuing

"In this world, visa time is life.

"Quantifying the value of life to earn visa time is essentially trading life for life.

"As for whether this exchange is worthwhile, one must consider the specific risks, rewards, and probabilities.

"Of course, this brings us to another trap in this game: the 'Chip Exchange Machine.'

"From the very beginning, players are thrown into a gambling match against the Chip Exchange Machine, and the rules clearly state that this is an unfair game.

"The Chip Exchange Machine can draw an extra card, choosing three out of four to form the best possible hand. Furthermore, it is always the dealer and always bets first.

"Generally speaking, betting first puts you at a disadvantage. However, because the Chip Exchange Machine draws an extra card, its hands are naturally stronger, which in turn increases our cost for calling and seeing the cards. If you want to see the cards, you must endure an additional loss of 500 chips.

"Most players lose everything after just two rounds.

"They either fold out of fear before calling, or they call and still lose, incurring even greater losses.

"This severe negative feedback leads most people to naturally assume that there is no way to win against the Chip Exchange Machine and that playing against it is a pure waste of chips.

"This is the greatest information barrier in the game.

"This barrier even kept smart people like Cai Zhiyuan and Officer Cao on the outside."

Fu Chen nodded thoughtfully after a moment. "Yes, the four of us played a total of twenty-four rounds against the Chip Exchange Machine.

"We lost every single one of the first few rounds.

"The Chip Exchange Machine has nearly a 50% chance of getting a pair; even if it doesn't get a pair, it inevitably gets high-card hands of Jack or higher.

"Later, I gave up. I would just fold without even looking at my cards, because looking at the cards would cost another 500 chips anyway."

He looked at Cai Zhiyuan. "However, we did discuss the relevant probabilities at the time, and Cai Zhiyuan did some rough calculations.

"But whether it was the calculated probability or the perceived probability during the game, the win rate was extremely low."

As a programmer, Cai Zhiyuan was the most proficient in probability.

And poker-style games like this are strongly correlated with probability.

Because the probability of every type of hand can be precisely calculated, and the difference in probability between two types of hands naturally determines the win rate.

Originally, many people thought Cai Zhiyuan would surely be the best at these kinds of games.

They didn't expect that his performance would actually be nothing special.

Wang Yongxin, a boss who didn't know much about probability, and Lin Sizhi, a liberal arts student, ended up being the only two people in Community 17 to discover the loophole in the Chip Exchange Machine.

Cai Zhiyuan sighed silently. "I did calculate the relevant probabilities.

"But the more I calculated, the more I ended up misleading myself."

Wang Yongxin suggested, "Why don't we take this opportunity to calculate the probabilities thoroughly? Even though the game is over, at least everyone can understand it clearly, regardless of whether they won or lost."

Cai Zhiyuan nodded. "Alright."

He took a pen and paper and began calculating rapidly.

The initial calculations weren't too complex, but when it came to calculating the changes in probability for choosing three out of four cards, the workload increased exponentially.

Although Cai Zhiyuan calculated quickly, it still took him several sheets of paper to provide the final answer.

"It should be more or less correct."

Cai Zhiyuan handed the paper covered in numbers to Wang Yongxin.

Wang Yongxin read it from beginning to end, his face showing expressions of considerable surprise from time to time.

Then, he looked at the group. "Let me pose a question first: in this game, what kind of hand do you think is worth placing a heavy bet on?"

The group looked at each other, speculating.

"A straight? A straight is a medium-sized hand."

"I don't think so. Isn't a straight already a low-probability hand? I think a pair is enough."

"A pair of 10 or higher? That should be about it."

"Wait, first, let's clarify what 'placing a heavy bet' means. If you bet all your remaining chips, would you dare bet on a pair to win the whole pot? I think it has to be at least a flush."

Clearly, even in this situation, they could not reach a consensus.

Because everyone's risk tolerance and estimation of various hand probabilities differed; when these two factors overlapped, the decisions made were naturally vastly different.

Wang Yongxin provided the answer: "Out of 52 cards, there are 22,100 possible combinations when picking three cards.

"Among them, there are 52 ways to get Three of a Kind, a probability of 0.24%;

"There are 48 ways to get a Straight Flush, a probability of 0.22%;

"There are 1,096 ways to get a Flush, a probability of 4.96%;

"There are 720 ways to get a Straight, a probability of 3.26%;

"There are 3,744 ways to get a Pair, a probability of 16.94%;

"And there are 16,440 ways to get High Card, a probability of 74.39%."

As soon as he said this, everyone was shocked.

"What? The probability of a pair is only 17%?"

"The probability of a Straight Flush is lower than Three of a Kind? The probability of a Straight is lower than a Flush?"

"The probability of High Card is 74.39%? Doesn't that mean if you get a pair, even the smallest pair, you have nearly a 75% win rate in a head-to-head match?"

Clearly, this probability completely overturned conventional intuition to some extent.

Everyone knew that High Card was the lowest, and that Straights through Three of a Kind were big hands, but how high was the actual win rate? Few had ever truly calculated it.

Most people estimated that the win rate for High Card might be around 50%, and even a pair might not reach 60%.

But in reality, even the smallest pair had a win rate as high as 75%, and the win rate starting from a Straight was as high as over 90%.

Wang Yongxin continued, "What do you all think the win rate is for the Chip Exchange Machine after it draws an extra card and forms a hand from four cards choosing three?

"Forget it, I won't keep you in suspense. I'll just give you the results.

"The probability of Three of a Kind is 0.9%;

"The probability of a Straight Flush is 0.85%;

"The probability of a Flush is 16.8%;

"The probability of a Straight is 8.4%;

"The probability of a Pair is 27.3%;

"The probability of High Card is 45.60%.

"The advantage of drawing an extra card is indeed massive. Even if both sides have High Card, the Chip Exchange Machine will most likely optimize it into a larger High Card.

"Therefore, in the actual process of gambling against the Chip Exchange Machine, losing is indeed the high-probability event."

Cai Zhiyuan nodded. "Yes, I only calculated up to that step in the game and didn't go any further against the machine.

"However, if you know that the 'Chip Exchange Machine' will never fold, the strategy changes completely.

"Although the win rate looks low, during a match, once we get a relatively large hand, we can obtain doubled returns through continuous betting.

"Suppose we get a large pair, like Kings; then, when facing the Chip Exchange Machine, we actually have a win rate of over 70%.

"Suppose we get a Flush, the win rate is even higher than 80%."

Jiang He frowned slightly. "But those kinds of hands are very hard to get, aren't they?"

Wang Yongxin nodded. "It is hard to get them, but the Chip Exchange Machine actually provides many opportunities.

"For every 100ML of blood drawn, you can gamble against the Chip Exchange Machine for 1 to 20 rounds