Independence means that one event, random quantity, or error does not depend on another. In repeated trials, earlier outcomes do not change the probability of later outcomes. section 2.1 section 3.3.2
Use independence to choose the appropriate probability rule and interpret model assumptions:
- Apply the multiplication rule when independent events occur together or sequentially; multiply their individual probabilities. section 3.2 section 2.5
- Do not use conditional probabilities for independent events; reserve them for dependent events. section 3.3.2
- Distinguish independence from mutually exclusive events: mutual exclusivity supports the addition rule, whereas independence supports multiplication. section 2.2.2 section 2.5
Repeated trials and sampling with replacement
After a fair coin lands heads, the next flip still has the same probability of heads. Likewise, after five successive heads, the next flip is not made less likely to be heads. Returning a selected queen of diamonds to a 52-card deck preserves the probability of selecting it again at 1 in 52. section 2.1 section 3.3.2
Independence also functions as a linear-regression assumption: residuals should be independent, alongside linearity, homoscedasticity, and normality. Testing these assumptions supports reliable interpretation and decision-making. section 4.3
