"Do you think friendship can be explained with probability?"
Riku suddenly brought this up. Afternoon light streaming through the club room window illuminated specks of dust.
"You're thinking strange things again," Yuki chuckled wryly.
"But it's an interesting question," Aoi opened a notebook. "Human relationships are filled with uncertainty. Probabilistic reasoning might be applicable."
"Probabilistic reasoning?"
"Also called Bayesian thinking. It's an approach where you update your beliefs each time you gain new information."
Riku leaned forward. "For example, I talk to Yuki. If the response is good, does the probability of becoming friends increase?"
"Exactly. You calculate the posterior probability based on observed data from the prior probability."
Aoi drew a diagram on the whiteboard.
"Initially, assume the probability of becoming friends is 50 percent. This is the prior probability. Next, we observe that Yuki responded with a smile. Under this condition, what's the probability of becoming friends?"
Yuki pondered. "Smiling in response often happens when someone wants to be friends, but it could also be out of politeness..."
"Precisely. To find P(friends|smile), we use Bayes' theorem."
Aoi wrote the equation.
"P(friends|smile) = P(smile|friends) × P(friends) / P(smile)"
"I understand the numerator, but the denominator?"
"The overall probability of smiling. It's the total including both wanting to be friends and being polite."
Riku wrote in his notebook. "So if we increase observations, the probability becomes more precise?"
"Yes. Exchanging messages multiple times, eating lunch together. Each new piece of evidence updates the posterior probability."
"Conversely, with few observations?"
"Uncertainty remains high. That's why gathering information is important."
Yuki suddenly thought of something. "But what if we misinterpret?"
"Sharp observation. Even if the observation itself is correct, if the estimate of P(smile|friends) is wrong, the conclusion goes awry too."
"It's a problem of subjective probability," Aoi supplemented. "Even with the same data, the posterior probability changes depending on the interpreter's experience and beliefs."
Riku laughed. "So that's why when you and I look at the same person, our judgments about whether we could become friends differ."
"Exactly. Probabilistic reasoning doesn't promise complete objectivity. But it provides a framework for updating beliefs based on data."
Yuki looked back at the notebook. "So can the probability of friendship reach zero?"
"Theoretically yes. But as long as the probability isn't zero, possibility remains."
"You mean there's hope?" Riku said.
"You could say that. In Bayesian terms, you can always update when new evidence appears."
Aoi quietly added. "That's why information theory can apply to human relationships too. It's a tool for handling uncertainty."
Yuki smiled. "What was the probability of meeting you both?"
"Maybe it was low initially. But the moment you chose to come to the club room, the posterior probability shot up."
"Choices change probabilities..."
Riku gazed at the whiteboard. "Probabilistic friendship theory is surprisingly deep."
"Friendship can't be fully measured by probability," Aoi said. "But the process of building relationships amid uncertainty can be understood probabilistically."
"In the end, what matters is taking action," Yuki concluded.
"Action generates observational data. And updates probabilities," Aoi nodded.
Outside the window, voices from other club activities echoed. Perhaps each person was building human relationships while making their own probabilistic inferences.
"Who should we talk to next?" Riku laughed.
"Before calculating probabilities, first take action," Yuki stood up.
Bayes' theorem continues working quietly today between people.