Random and Non-Random Patterns

Order out of chaos

• A single random outcome is unpredictable — but many of them form a stable pattern.
• Randomness 随机性 means individual results vary and can't be foreseen one at a time.
• Yet over many repetitions, the proportions settle down to something predictable.
• Probability is the math of that long-run stability.

Could it be just chance?

• The core question: is a result a real effect, or could it happen by chance 偶然?
• Random variation alone can produce streaks and clusters that look meaningful.
• To judge "surprising," we compare what we saw against what chance would do.
• If chance can easily produce it, it isn't strong evidence of anything.

Random in the short run, predictable in the long run: the proportion of heads settles near 0.5

Probability = long-run frequency

• Probability 概率 is the long-run relative frequency 长期相对频率 of an outcome.
• $P(\text{heads}) = 0.5$ means: over many tosses, about half land heads.
• It's not a promise about the next toss — it's about the long run.
• A probability is always a number between $0$ and $1$.

Why we need a model

• To decide if data are surprising, we need a probability model to compare against.
• The model says what outcomes chance should produce, and how often.
• Then "surprising" = far from what the model predicts.
• Units 4–5 build these models so later units can test claims.

Write from long-run relative frequency

• A spinner is spun thousands of times.
• The proportion of red settles near one value.