"Aoi-senpai, I came on time today!"
Riku said proudly. Yuki and Aoi looked at each other in surprise.
"Really?" Yuki checked the clock. "Exactly 3 PM... it's true."
"Amazing information content," Aoi laughed.
"Information content?" Riku tilted his head.
"Yes. The more unpredictable an event, the higher its information content. This is called self-information."
Aoi wrote an equation in the notebook. "I(x) = -log₂ p(x)"
"Information content of event x occurring with probability p(x). The lower the probability, the higher the information content."
Yuki started calculating. "If the probability of Riku coming on time is 0.1..."
"-log₂(0.1) ≈ 3.32 bits," Aoi answered.
"Is that a lot?" Riku asked.
"Very much. For example, a fair coin toss result is 1 bit. But Riku's on-time arrival is 3.32 bits. More than three times the surprise."
"Am I that unpredictable?"
"Not in a bad way," Yuki laughed. "Today it's a positive surprise."
Aoi continued. "In information theory, surprise can be quantified. Rare events have high informational value."
"So because I'm usually late but came on time, today has high information content?"
"Exactly. If Riku were always on time, today's arrival would have nearly zero information content."
Riku pondered. "Information has no value in ordinary things."
"In a sense. Completely predictable events yield no new information."
Yuki thought of an example. "So 'the sun rose in the east' has zero information content?"
"Nearly zero. Probability is close to 1," Aoi nodded. "But 'the sun rose in the west' would have information content approaching infinity."
"Infinity?" Riku was surprised.
"Because it's an event with nearly zero probability. log₂(0) diverges."
"But that wouldn't actually happen, right?" Yuki said.
"Right. So information theory considers the range of possible events. Events with zero probability aren't handled."
Riku took out his smartphone. "News always reports unusual things."
"Exactly from an information content perspective," Aoi acknowledged. "News like 'another peaceful day' has low information content. But 'a meteorite fell' is high."
"That's why it has news value," Yuki summarized.
"Exactly. People pay attention to events with high self-information. It was evolutionarily important."
Riku's face became serious. "Should I come on time every day? Or is it more valuable to come occasionally?"
Aoi laughed. "An ontological question. But information-theoretically, moderate unpredictability might be optimal."
"Moderate?"
"If completely random, patterns can't be learned. If completely predictable, there's no new information. The middle is good."
Yuki looked convinced. "Like music. There's repetition, but sometimes unexpected developments."
"Perfect metaphor. Moderate surprise makes information valuable."
Riku suddenly thought of something. "Then next time, I'll come an hour early!"
"That also has high information content," Aoi smiled wryly. "But usefulness is a separate issue."
"Usefulness?"
"Even if information content is high, it's meaningless if not useful. Information value depends on context."
Yuki laughed. "Like Riku coming too early and no one's there."
"That's noise," Riku admitted.
"No," Aoi corrected. "Not noise, but wrong information. High information content, but different from expectations."
The three laughed. High information content and usefulness are different things. But occasional surprises add color to daily life.
"Next time, I'll just come on time normally," Riku declared.
"That might have the lowest information content," Aoi laughed.
But that's fine. In predictable daily life, occasionally mixed high-information events. That might be the optimal balance that information theory shows.