"Do you think coincidences exist?"
Mira spoke up, which was unusual.
Yuki turned around in surprise. "Coincidences?"
"Meetings, timing, things like that."
Aoi responded with interest. "In probability theory, everything is a probabilistic event."
Riku laughed. "Love is also probability?"
"In a sense," Aoi opened the notebook. "The probability of two people meeting, the probability of talking, the probability of developing affection."
"But," Yuki thought, "those aren't independent, right?"
"Sharp. That's the difference between independent and dependent events."
Aoi wrote on the whiteboard. "If P(A∩B) = P(A)・P(B), they're independent. But often, P(A∩B) ≠ P(A)・P(B)."
"Meaning?" Riku asked.
"The probability of meeting and the probability of developing affection aren't independent. Meeting influences the probability of affection."
Mira said quietly, "Conditional probability."
"Yes. Conditional probability. P(affection|meeting) differs from P(affection)."
Yuki understood. "When the condition of having met exists, the probability of affection changes."
"Exactly. So love is a probabilistic chain."
Riku pondered. "So the first meeting is most important?"
"Statistically, perhaps. Initial conditions influence subsequent probabilities."
Aoi gave an example. "Like a Markov chain. The current state determines the probability of the next state."
"Markov chain?" Yuki asked.
"A type of stochastic process. The future depends only on the present, not the entire past."
Mira supplemented. "First-order Markov. Simple but powerful."
"Love is a Markov chain?" Riku laughed.
"Not a perfect model, but interesting as an approximation."
Aoi continued. "From the 'met' state to the 'talked' state. From 'talked' to 'became friends.' Each transition has a probability."
Yuki drew a diagram in the notebook. A state transition diagram.
"But," Mira said, "Love is not Markovian."
"What do you mean?" Yuki asked.
"All past experiences influence. The impression of the first meeting, all conversations, all moments."
"I see," Aoi acknowledged. "Human memory breaks perfect Markovian property."
Riku raised his hand. "So love is unpredictable?"
"Can be treated probabilistically, but too complex for accurate prediction."
Aoi offered another perspective. "But that's what makes it interesting. If it were completely predictable?"
"Boring," Yuki answered immediately.
"Yes. Uncertainty makes love dramatic."
Mira smiled. "Entropy of love is high."
"The entropy of love is high," Aoi translated. "A state with many possibilities."
Riku looked out the window. "Me coming here today is also a probabilistic event?"
"Yes. Not probability 1. If something had been different, you might not have come."
"Then us meeting too..."
"A chain of probabilistic events," Aoi said quietly. "But that probability wasn't zero."
Yuki thought. "All the probabilistic events in the universe gathered us here."
"Poetic," Aoi smiled. "But not wrong."
Mira stood up and wrote on the whiteboard.
"P(we meet) = Π P(events)"
"The product of probabilities of all events," Aoi supplemented. "If even one were zero, we wouldn't have met."
Riku's face became serious. "So all meetings are miracles?"
"In probability theory, realizations of rare events."
"But," Yuki added, "that's why we should cherish them."
Aoi nodded. "Lower probability events have higher information content. Our meeting also carries high information."
Mira said quietly, "Cherish the improbable."
The four quietly savored their probabilistic meeting. In the universe's probability distribution, they met. And now, they are here. It was a small miracle carried by probabilistic events.