"Probability isn't prophecy, is it?"
Yuki murmured while reviewing the lesson.
"Of course not," Aoi answered. "Probability only shows long-term trends. Individual results can't be predicted."
Riku looked dissatisfied. "Then it's meaningless."
"Not at all," Aoi countered. "There's a concept called expected value."
Yuki opened the notebook. "Expected value is average, right?"
"Strictly, probability-weighted average. But it has deeper meaning."
Aoi took out a die.
"Roll a die. If 1 comes up, you get 100 yen, otherwise nothing. Is this game worth participating in?"
Riku thought. "1/6 probability for 100 yen, so expected value is about 17 yen?"
"Correct. So if participation fee is 17 yen or less, long-term you profit."
"But just once, you might lose," Yuki pointed out.
"Right. That's the difference between probability and expected value. One result is uncertain, but with many repetitions, it approaches expected value."
Aoi continued. "Law of large numbers. As trial count increases, average converges to expected value."
Riku suddenly thought. "Should life choices also be considered by expected value?"
Aoi answered carefully. "It's one perspective. But variance should also be considered."
"Variance?"
"Result dispersion. Even with same expected value, risk can differ."
Aoi wrote two choices.
"A: Certainly get 500,000 yen B: 50% probability 1,000,000 yen, 50% probability 0 yen"
"Expected value is 500,000 yen for both. But B has higher risk."
Yuki pondered. "Which to choose depends on personality?"
"Yes. Risk-averse people choose A, risk-neutral people see them as equal, risk-loving people choose B."
Riku raised his hand. "I'd choose B. I like gambling."
"That's your utility function," Aoi laughed. "Not just amount, but subjective value matters."
Yuki asked. "Is this related to information theory?"
"Deeply related," Aoi became serious. "Entropy is expected information content. Calculated from probability distribution."
"Ah, it connects."
"Yes. H(X) = E[-log P(X)]. Measuring information content in expected value framework."
Riku pondered. "So higher uncertainty means higher expected information content?"
"Exactly. Uniform distribution has maximum entropy. All results are equally uncertain."
Aoi drew a diagram on the whiteboard.
"Probability theory is a framework for mathematically handling uncertainty. Information theory is built on top of it."
Yuki summarized in the notebook. "Probability isn't prophecy, but a tool to quantify uncertainty."
"Perfect."
Riku laughed. "Life is also a random variable. Results are unknown, but expected value can be calculated."
"Philosophical," Yuki said.
"But life is once only," Aoi said quietly. "Law of large numbers doesn't apply. So deciding by expected value alone is dangerous."
"Then what should we do?"
"Know your own values. What you cherish. That's your utility function."
Riku nodded. "Expected value matters, but so do your feelings."
"Yes. Probability theory is a tool, not an answer."
Yuki smiled. "Probability lessons might be life's miniature."
"Good insight," Aoi acknowledged.
The sunset illuminated the classroom. Probability and life, uncertainty and choice. It was a moment where they intersected.