The shared language of uncertainty

Conditional probability shrinks the sample space to the fact that B happened. Bayes theorem updates a prior with evidence into a posterior, and it explains why a positive test for a rare disease is still unlikely.

As you draw more samples, the average settles toward the expected value (the law of large numbers). Monte Carlo uses random samples to estimate areas or integrals this way, with error shrinking like one over the square root of n.

The mean computed from each sample varies, and that spread is the sampling distribution. The standard error of the sample mean is σ/√n, so it narrows as the sample grows. Thanks to the central limit theorem, the sample mean becomes bell-shaped even when the population is not normal.

Add a margin of error to a point estimate and you get a confidence interval. Ninety-five percent confidence means the procedure captures the true value in 95 percent of repeated samples. Higher confidence widens the interval, a larger sample narrows it.

Assume the null hypothesis is true and measure how surprising the data is with a p-value. Reject the null when the p-value falls below the significance level. A Type I error is a false alarm, a Type II error is a miss, and power is the chance of not missing.

The correlation coefficient measures the strength of a linear relationship, and least squares finds the line that minimizes the sum of squared residuals. R-squared is the fraction of variance explained. As noise grows, correlation drops toward zero and the confidence interval on the slope widens.