"Do you think Riku will be late today?"
Yuki asked Aoi. They were waiting for Riku in the club room.
"90 percent probability of being late," Aoi answered immediately.
"But yesterday he said, 'I absolutely won't be late tomorrow.'"
"That's included in the 90 percent."
As expected, Riku burst in 15 minutes late.
"Sorry! But today there's a reason. Mira collapsed in the hallway..."
Behind him, Mira quietly entered. She showed no sign of having collapsed.
"Riku, was that a lie?" Yuki was exasperated.
"No, seriously! But she stood up right away..."
Mira nodded slightly. Apparently it was true.
Aoi opened the notebook. "Interesting situation. There might be correlation between Riku's lateness and Mira's condition."
"Correlation?" Yuki asked back.
"Whether two events are related to each other. Mutual information quantifies this."
Aoi wrote an equation. "I(X;Y) = H(X) - H(X|Y)"
"How much uncertainty in Y is reduced by knowing X."
Riku looked confused. "Meaning?"
"For example, let X be Riku's lateness, Y be Mira's condition. If they're completely independent, knowing X tells you nothing about Y. Mutual information is zero."
"But today, the reason I was late was Mira," Riku said.
"Yes. So there's some correlation. Mutual information has a positive value."
Mira quietly drew a diagram on the whiteboard. A Venn diagram with two overlapping circles.
"The overlap is mutual information," Aoi explained. "Information that X and Y share."
Yuki thought. "What if they're perfectly correlated?"
"Mutual information is maximized. X can completely predict Y."
Riku clapped his hands. "Then I should act more unpredictably!"
"Why?" Yuki asked.
"Because if you're predictable, it's boring, right?"
Aoi smiled. "Philosophical. But in information theory, predictability increases efficiency."
"How so?"
"For example, consider a weather prediction system. If yesterday's weather has strong correlation with today's, you can efficiently predict today from yesterday's data."
"So patterns make things easier," Yuki summarized.
"Exactly. But Riku has a point too. A perfectly predictable world has zero new information."
Mira showed a note. "Surprise = Information"
"Yes. Unpredictable events have high information content."
Riku became proud. "So my lateness has informational value!"
"That's different," Aoi corrected immediately. "Being unpredictable and being chaotic are separate. Good unpredictability has structure with surprises."
Yuki thought of an example. "Like music? It has patterns but sometimes unexpected developments."
"Perfect example. Completely random sound only sounds like noise. But appropriate surprise within structure becomes beautiful music."
Riku's face became serious. "Should I come on time sometimes then?"
"That would have higher information content," Aoi laughed.
Mira drew a new diagram. A time-series graph. Riku's lateness record.
"Almost all late!" Riku exclaimed.
"So predictable. From a mutual information perspective, checking the clock tells you Riku's location."
Yuki laughed. "So Riku's actions and time have strong correlation."
"Negative correlation though," Aoi supplemented. "The closer to start time, the farther away Riku is."
Riku looked frustrated. "Tomorrow, I'll really do something unpredictable..."
"Come on time?" Yuki hoped.
"No, come 30 minutes early!"
Aoi and Mira laughed simultaneously. It was rare.
"That's predictable in a different way," Aoi said.
Laughter echoed through the club room. In predictable daily life, occasionally appearing unpredictable moments. That is information, and value.