"I wonder how many points I'll get on tomorrow's test."
Riku murmured anxiously.
"Let's think probabilistically," Aoi suggested.
"Probability?" Yuki tilted her head.
"Test scores are random variables. There are multiple possibilities, each with an assigned probability."
Riku took out his notebook. "What does that mean?"
Aoi wrote on the whiteboard. "From 0 to 100 points, there are various results. From past data, we can estimate the probability distribution for each score."
"Last time I got 60 points," Riku said.
"That's one sample. We look at the distribution from multiple tests."
Yuki started calculating. "Riku's average over the past 5 tests is 65 points."
"That's the expected value," Aoi explained. "E[X] = Σ x・P(x). Multiply each result by its probability and sum them."
"If the expected value is 65, will I get 65 tomorrow too?" Riku held hope.
"Not necessarily," Aoi corrected. "Expected value is the average. Actual results vary up and down."
"Vary?"
"There's a concept called variance. Var(X) = E[(X - E[X])²]. The average of how far results deviate from the expected value."
Yuki looked at past data. "Riku's scores vary quite a bit. From 40 to 80 points."
"The variance is large," Aoi confirmed. "Meaning prediction uncertainty is high."
Riku felt discouraged. "So I don't know what score I'll get?"
"Not completely. But probability distribution becomes a prediction tool."
Aoi drew a diagram. A curve like a normal distribution.
"If scores follow a normal distribution, they concentrate around the expected value. With standard deviation σ, about 68 percent fall within E[X]±σ."
Yuki calculated. "Riku's σ is about 15. So tomorrow's score will likely be between 50 and 80 points."
"I don't want 50 points," Riku muttered.
"But," Aoi encouraged, "if you study today, you can shift the distribution."
"Shift the distribution?"
"Update the prior probability to posterior probability. The action of studying affects tomorrow's probability distribution."
Yuki became interested. "Like Bayesian inference."
"Exactly. New information—today's studying—changes the future prediction."
Riku became serious. "So if I study now, the expected value rises?"
"It's highly likely to rise. If we know the correlation between study time and scores from past data, we can predict more accurately."
Aoi drew another diagram. "If study time is x, then score = a・x + b + noise. A linear model."
"Noise?"
"Perfect prediction is impossible. Physical condition, problem difficulty, luck. These are added as noise."
Yuki said, "But on average, studying raises scores."
"Accurate," Aoi nodded. "There's noise, but we can capture the trend."
Riku opened his planner. "If I study 3 hours today, what happens to the expected value?"
Aoi calculated. "From past data, assuming about 5 points increase per hour, the expected value becomes 80 points."
"80 points!" Riku rejoiced.
"But that's the expected value. Actually it will vary between 70 and 90 points."
"That's still enough." Riku stood up. "I'll study now."
Yuki laughed. "Probability moved Riku."
Aoi supplemented. "Probability theory guides action. We can't predict perfectly, but it helps rational judgment."
"But," Yuki asked, "if tomorrow's score is 60, is the model wrong?"
"No," Aoi explained. "Probabilistic predictions can't be evaluated from one result. What matters is whether the predicted distribution matches the actual distribution in the long term."
"So even if it's lower than expected occasionally, that's within the range of probability variance?"
"Yes. So there's no need to be overly emotional about single results. In the long view, it converges to the expected value."
Riku took out his textbook. "Got it. I'll work hard today and raise the long-term expected value."
"Good strategy," Aoi acknowledged. "Not being swayed by short-term fluctuations, but seeing long-term trends. That's the wisdom of probability theory."
The three discussed waves of probability in the sunset club room.
The future is uncertain, but probability distributions become a compass.
Even if perfect prediction is impossible, better choices are possible.