Kakegurui: Ten Consecutive Guaranteed Hits

Volume 1 Chapter 45: The Dice's Expectation Problem

When Kiyohei heard this number, he thought he had misheard. One million yen? Even if you were to reopen a gambling establishment, you wouldn't need one million! Where did Mary get so much money from!?

"...Mary, where did you get one million yen?" Kiyohei looked at Mary coldly. "Did you win seven hundred and fifty thousand in just one morning?"

Tsuzura whispered, "Mary used the twenty-five thousand she won, and I had fifty thousand on hand."

Kiyohei glanced at her in surprise. Indeed, Tsuzura had fifty thousand. But wasn't that the money she had set aside as insurance for herself? She hadn't even been willing to use it when she was reduced to livestock; how could she easily give it to Mary now? Didn't she worry about what would happen if Mary lost?

Well... that was her choice, and Kiyohei had no right to interfere.

"What about the remaining twenty-five thousand?" Kiyohei continued to ask.

Tsuzura looked somewhat hesitant, as if it wasn't easy to explain. Meanwhile, Mary covered her face with her hands, unable to face Kiyohei, seemingly realizing how foolish she had been.

"The remaining twenty-five thousand is what I lent her," Sachiko said with amusement, enjoying the scene. She suddenly spoke up, "After all, a gambling game can't take place without different amounts of stakes, can it?"

"One million..." Kiyohei was nearly amused. "Mary, you only have this much money on hand, and you went all-in without hesitation? You even asked the Student Council for a loan? Have you considered the consequences if you lose?"

Kiyohei had specifically advised her that even if she wanted to make money through gambling, she should choose low-risk games, not high-stakes gambling that would leave her with no way out if she lost.

Mary, who had promised herself not to gamble, had suddenly taken a seat at the table for a one-million-yen game?

"As long as I win, it'll be fine, right? Trust me, I know the rules of this game now. Believe me, I can win..." Mary tried to reassure Kiyohei, despite her exasperation.

"Who cares whether you win or lose?" Kiyohei rolled her eyes in frustration.

Gambling, like "betting one's life," was one of those things where once you made quick money, you wouldn't want to earn it slowly and steadily. You'd start looking for shortcuts.

Just like stock trading or drug addiction, gambling was an abyss. Once you started, you'd gradually be consumed by it, with no way to escape. To break free from gambling, you had to avoid getting deeply involved in the first place.

Kiyohei didn't want to see Mary fall into this abyss, especially since she was borrowing money to gamble. It was one thing if she had used her own money, but to borrow from the Student Council...

Kiyohei was at a loss for words.

The match was being supervised by Student Council members, and even if Mary regretted it now, it was probably too late.

"Never mind, it's up to you to handle it." Kiyohei no longer cared about her.

One million yen was equivalent to over fifty thousand Chinese yuan, which was not a small amount. If she won, everyone would be happy, but if she lost, Mary would have to learn a lesson from this failure and understand the dangers of gambling.

Mary also knew that Kiyohei was looking out for her. Seeing that Kiyohei had stopped objecting, she breathed a sigh of relief and responded with a bright smile, "Trust me, I can win!"

Kiyohei and Runa arrived just in time, before the gambling had begun. Although Mikura didn't know why Kiyohei rubbed her the wrong way, at Sachiko's request, she explained the rules to the two.

They were about to play a game called "Three Hit Dice," and the rules were as simple as the name. The game was conducted by the dealer, Mikura, who would roll the dice.

In this game, one, two, and three were considered "Down," while four, five, and six were considered "Up." The two players had to guess whether the dice would land on Up or Down for three consecutive rolls.

Both sides would write their guessed results on paper and then hand them to the dealer, Mikura.

She would continue to roll the dice until one of the guessed results appeared first.

The rules of this game were not complicated, and for easier understanding, let's use a coin toss as an example.

In simple terms, it was like guessing the outcome of a coin toss, where heads were considered "Up" and tails were "Down."

For instance, Mary guessed "UUD," which means "Up Up Down." If Mikura, the dealer, rolled the coin three times in a row with heads (Up), and then on the fourth roll got a tails (Down), which matched Mary's guess, then Mary would win the round.

If it didn't match, Mikura would continue to roll until one side's prediction came true.

Whether it's a coin or a die, assuming no manipulation, the probability of getting heads or tails, or in this case, Up or Down, is the same.

The probability of getting three heads in a row or three tails in a row is both 1/8, as there are eight possible outcomes in total.

In classical probability theory, when there is no evidence to prove that the probability of one event is greater than the probability of another, you can consider them equally likely.

However, in real-life situations like this gambling game, it can't be explained using traditional probability definitions. There are too many factors at play, such as whether the coin or die is "perfect," whether the dice are evenly manufactured, whether their center of gravity is in the exact center, whether the roulette wheel favors a particular number, and even the dealer's actions, which might affect the centering of the dice or coin.

So, if you simply think of this as a probability calculation, you're mistaken. It's actually a classic mathematical expectation problem within probability theory.

In Kiyohei's previous life in college, she often solved these types of problems. When she heard Mikura explain the rules, she began to calculate the expected values in her mind.

There are various methods for calculating expected values, but let's skip that and go straight to the results.

The expected values for the two possibilities, "Up Up Up" and "Down Down Down," are both 14.

For the two possibilities, "Up Down Up" and "Down Up Down," the expected values are both 10.

And for the four possibilities, "Up Down Down," "Down Up Up," "Up Up Down," and "Down Down Up," the expected values are all 8.

In other words, the four arrangements where the opposite outcome occurs first have a higher likelihood of appearing. So, if you have to guess, you can write down any of those four arrangements.

Of course, even if the probability of other arrangements appearing is smaller, there's still a possibility they could happen.

In the end, it's still a game of luck.