A probability model lists the possible outcomes and gives each one a probability. Some models come from equally likely outcomes; others come from experiments. Comparing a model with real results tells you whether to trust it.
Key idea
In a uniform model, every outcome has the same probability. In other models, probabilities differ, and they can be estimated from data. All the probabilities in a model add to 1. A model predicts about (probability × trials) occurrences; results far from that, over many trials, suggest the model is wrong.
Why it works
A model is a claim about how chance behaves. Small differences between the prediction and the data are normal, because results vary. But if a spinner with 4 "equal" parts lands on red 180 times in 400 spins instead of about 100, chance alone almost never does that: the parts are probably not equal.
Worked examples
Worked example: Does it fit?
A spinner has 4 equal parts, one red. In 400 spins, it lands on red 103 times. Does the data fit the model?
- The model predicts $\frac{1}{4} \times 400 = 100$ reds.
- 103 is close to 100, a normal amount of variation. The data fit the model.
Answer: Yes
Worked example: It does not fit
Another "equal parts" spinner lands on red 180 times in 400 spins. Does it fit?
- Again the model predicts about 100.
- 180 is far from 100 after so many spins, so the model does not fit. The red part is probably bigger than a quarter.
Answer: No
Worked example: A model from data
A thumbtack lands point up 62 times in 100 drops. Make a model and predict the number of point-up landings in 500 drops.
- The outcomes are not equally likely, so use the data: $P(\text{up}) \approx \frac{62}{100} = 0.62$ and $P(\text{down}) \approx 0.38$. They add to 1 ✓.
- Prediction: $0.62 \times 500 = 310$.
Answer: About 310