Random and Non-Random Patterns
Introduced| English |
|---|
| randomness/ˈrændəmnəs/ |
| probability/ˌprɒbəˈbɪlɪti/ |
| chance/tʃæns/ |
| long-run relative frequency/lɒŋ rʌn ˈrelətɪv ˈfriːkwənsi/ |
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.
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.
Probability describes the long run, not the next trial. "$P=0.5$" does not mean heads and tails must alternate, or that a run of $5$ heads is "due" to end — each toss is fresh. The law of averages ("I'm due for a win") is a fallacy; chance has no memory. The stability appears only over many repetitions.
Flip a fair coin.
- Short run: $5$ flips could easily be HHHHH — chance is streaky.
- Long run: over $10{,}000$ flips, the fraction of heads is very close to $0.5$.
- So $P(\text{heads}) = 0.5$ is a statement about the long run, not any single flip.
Randomness makes individual outcomes vary unpredictably, but over many repetitions the relative frequency stabilizes. Probability is that long-run relative frequency, always between $0$ and $1$. We need a probability model to judge whether an observed result is surprising or just chance.
A probability from 0 to 1
Probability lives on a scale from 0 (impossible) to 1 (certain).
Probability is best described as the long-run...
Probability = long-run relative frequency over many repetitions.
After 5 heads in a row on a fair coin, tails is 'due' and more likely on the next flip.
Each flip is independent — chance has no memory. That's the law-of-averages fallacy.
What is the largest value a probability can take?
Probabilities range from 0 to 1.
The property that individual outcomes vary and can't be predicted one at a time is called ___.
Randomness produces short-run variability but long-run stability.
Why do we need a probability model?
The model says what chance should do, so we can spot the surprising.