The Probability of Hand Types

Wang Yongxin paused slightly and continued, "In this world, visa time is life.

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

"As for whether this exchange is worthwhile, we still need to consider the specific risks, returns, and probabilities.

"Of course, this is where we need to mention another trap in this game: the 'Chip Exchange Machine'.

"Players are thrown into a gamble with the Chip Exchange Machine from the start, and the rules clearly state that it's an unfair gamble.

"The Chip Exchange Machine can draw an extra card and choose three out of four cards to form the best hand. It's always the dealer and always bets first.

"Generally, betting first puts you at a disadvantage, but because the Chip Exchange Machine can draw an extra card, its hand is naturally stronger. This increases our cost to call and open the hand. Anyone wanting to open the hand has to bear an additional loss of 500 chips.

"Most players, after playing a couple of rounds, basically lost everything.

"Either they folded without daring to call, or they called and still lost, incurring greater losses.

"The severe negative feedback made most people naturally believe that playing against the Chip Exchange Machine was unwinnable, a pure waste of chips.

"This is the biggest information barrier in the game.

"This barrier even kept out smart people like Cai Zhiyuan and Officer Cao."

Fu Chen pondered for a moment and nodded, "Yes, the four of us gambled against the Chip Exchange Machine a total of twenty-four times.

"In the first few rounds, we lost all of them.

"The Chip Exchange Machine had almost a 50% chance of getting a Pair, and even if it didn't get a Pair, it would definitely get a J or higher mixed hand.

"Later, I gave up. I would just fold without looking at the cards, as looking at them would cost another 500 chips."

He looked at Cai Zhiyuan, "However, we did discuss the relevant probabilities at the time. Cai Zhiyuan even did a rough calculation.

"But whether it was the calculated probability or the probability felt in the game, the win rate was very low."

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

And this kind of poker game was strongly related to probability.

Because the probability of each hand could be precisely calculated, and the difference in probability between two hands naturally determined the win rate.

Originally, many people thought Cai Zhiyuan would definitely be the best at this type of game.

Unexpectedly, his performance wasn't particularly outstanding.

Boss Wang Yongxin, who didn't understand probability well, and Lin Sizhi, a liberal arts student, were the only two people in Community No. 17 who discovered the loophole in the Chip Exchange Machine.

Cai Zhiyuan sighed softly, "I did calculate the relevant probabilities.

"But by calculating it again and again, I ended up misleading myself."

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

Cai Zhiyuan nodded, "Okay."

He picked up paper and pen and began calculating rapidly.

The initial calculation wasn't too complex, but when calculating the probability changes for choosing three out of four, the workload suddenly increased.

Although Cai Zhiyuan calculated very quickly, he still used several sheets of paper before arriving at the final answer.

"It should be correct now."

Cai Zhiyuan handed the paper filled with numbers to Wang Yongxin.

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

Then, he looked at everyone, "Let me keep you in suspense for a moment. In this game, what kind of hand do you think is worth betting heavily on?"

The crowd looked at each other and began to guess.

"A Straight, perhaps? A Straight is a medium-sized hand."

"That shouldn't be it. A Straight is already a low-probability hand, right? I think a Pair would be enough."

"A Pair of 10s or higher? That should be about right."

"Wait, let's clarify what 'betting heavily' means first. If you're betting all your remaining chips, would you dare to bet on a Pair to win the whole game? I think at least a Flush is needed."

Clearly, even in this situation, they couldn't reach a consensus.

Because everyone's risk tolerance was different, and their estimations of the probabilities of various hands were different. The combination of these two factors led to vastly different decisions.

Wang Yongxin gave the answer, "Taking any three cards from 52 cards, there are 22,100 possible combinations.

"Among them, there are 52 Leopard hands, with a probability of 0.24%;

"There are 48 Straight Flush hands, with a probability of 0.22%;

"There are 1,096 Flush hands, with a probability of 4.96%;

"There are 720 Straight hands, with a probability of 3.26%;

"There are 3,744 Pair hands, with a probability of 16.94%;

"And there are 16,440 mixed hands, with a probability of 74.39%."

As soon as these words were spoken, everyone was very surprised.

"Ah? 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 a high card is 74.39%? Doesn't that mean that as long as you get a pair, even the smallest pair, you have nearly a 75% chance of winning in a one-on-one?"

It was clear that this probability, to some extent, completely subverted conventional understanding.

Everyone knew that high card was the lowest, and straight to three of a kind were high hands, but how many people had actually calculated the specific win rates?

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

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

Wang Yongxin continued, "What do you think the win rate would be if the Chip Exchange Machine drew one more card, and then formed a hand from four cards chosen three?"

"Never mind, 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 a high card is 45.60%.

"The advantage of drawing an extra card is indeed significant. Even if both sides have high cards, the Chip Exchange Machine will most likely optimize for a higher high card.

"Therefore, in actual play against the Chip Exchange Machine, losing is indeed a high-probability event."

Cai Zhiyuan nodded, "Yes, I only calculated up to this point in the game and didn't continue to gamble against the Chip Exchange Machine.

"However, if you know that the 'Chip Exchange Machine' will not fold, the strategy changes completely.

"Although the win rate seems low, in a match, once we get a relatively high hand, we can double our returns by continuously raising the bet.

"Suppose we get a pair of Kings. Then, against the Chip Exchange Machine, we would actually have over a 70% win rate.

"Suppose we get a flush, we would have over an 80% win rate."

Jiang He frowned slightly, "But it's hard to get such hands, isn't it?"

Wang Yongxin nodded, "It is difficult 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. If you draw 600ML of blood, you can gamble up to 120 rounds.

"This number of rounds is actually enough to get many big hands. As long as you encounter one, your chips have a chance to double once.

"Not only that, if you can figure this out in advance, you can also fold immediately when you get a small hand, minimizing the game time.

Jiang He pondered for a moment, frowning, "But probability is often unreasonable, and small probability events are not entirely impossible.

"What if you encounter a bad beat? Wouldn't you lose all your capital?"

Wang Yongxin shook his head dismissively, "You can't think like that!

"If you are always worried about small probability events happening, you would fold every hand. How can you make money then?

"Besides, another advantage of gambling against the Chip Exchange Machine is that as long as you choose to call, you can reveal your cards at any time. Unlike gambling against players, if two people take turns raising, you can never reveal your cards.

"So, set your own stop-loss limit and adjust your bet amount in real-time based on the win rate.

"It's like doing business; it's a simple 'risk control' issue."

Everyone fell silent, clearly needing time to digest this information.

Fu Chen looked at the detailed rules on the big screen and sighed again.

The game didn't seem too complicated, but when broken down carefully, it had an unimaginable amount of detail.

"Alright, up to this point, we should have completely deconstructed this game.

"So... do you think this game really serves a screening purpose? In other words, do you think this way of allocating visa time is reasonable? Is it fair?"

Ding Wenqiang's face remained grim, "Unfair."

The others remained silent.

After a moment, Li Renshu said cautiously, "I believe it is unfair, but reasonable."