# 5 Basic probability

## Concepts

• Probability is the theory that allows us to make an inference from a sample to a population. It provides the mathematical and theoretical basis for quantifying uncertainty.
• Probability is also used directly to communicate risk. For example, what is the chance of survival 5 years after a heart transplant; what is the probability of conceiving within 3 cycles of IVF; what the probability of having the disease given a positive screening test.
• Independence is a fundamental concept in probability and statistics. It underlies the assumptions of many statistical approaches to inference eg, independent observations; the idea is also used more directly when we do tests for independence between two variables, eg; a chi-squared test of independence.
• Conditional probability is useful to understand, in part because it helps us understand independence, but also because many of the probabilities we use in statistical analysis are conditional, for example, a p-value is conditional on the null hypothesis being true.
• Conditional probabilities are also easy to mix up so understanding them is useful for general statistical literacy. Contingency tables and probability trees are useful tools to help us think clearly about conditional probabilities.

## Connections with other material

• The degree of uncertainty associated with any conclusions drawn from a sample is expressed through the language of probability.
• Independence is a key concept that arises frequently in statistical inference, understanding the probabilistic definition of independence will facilitate your understanding of later topics when this concept crops up.
• P-values are conditional probabilities; an appreciation of how conditional probabilities can be misinterpreted and muddled, will make you more statistically literate and give you a deeper understanding of p-values.