Why Probability Is Our Tool for Uncertainty: From Mars Life to Bayesian Logic

Probability and Uncertainty

How likely is life on Mars? What is the probability that an electron has a certain mass? What was the chance of sunny weather on July 9, 1816? These questions illustrate how we use probability to express uncertainty about events.

Although answers to binary questions like “Is there life on Mars?” are either true or false, we are interested in the probability of life based on current data and our understanding of Martian conditions. This probability depends on the information we possess, not on an inherent property of nature.

We use probability because we are uncertain about events, not because the events themselves are uncertain. This subjective definition of probability explains why the Bayesian school is often called subjective statistics. However, this does not imply that every proposition is equally meaningful; it merely acknowledges that our models are incomplete.

Understanding the world without models is impossible. Even if we try to escape social assumptions, our brains, shaped by evolution, are wired to think in probabilistic terms. Whether the universe is fundamentally deterministic or random, we employ probability as a tool to measure uncertainty.

Logic concerns effective inference. In classical Aristotelian logic a proposition is either true or false, whereas in the Bayesian view probability treats certainty as a special case: a true proposition has probability 1, a false one has probability 0.

Only when sufficient data confirm that life can exist and reproduce on Mars would we assign a probability of 1 to “Life exists on Mars.” Assigning probability 0 is difficult because unknown habitats or experimental errors may exist. This leads to Cromwell’s Rule , which advises against assigning probabilities of exactly 0 or 1 to logical statements.

Cox proved mathematically that incorporating uncertainty into logical inference requires probability theory, making Bayes’ theorem a logical consequence of probability. Thus Bayesian statistics extend logic to handle uncertainty without disparaging subjective reasoning.

Probabilities range between 0 and 1 and obey rules such as the multiplication rule: the probability of two events occurring together equals the probability of one event multiplied by the conditional probability of the second given the first.

Conditional probability is a cornerstone of statistics and essential for understanding Bayes’ theorem. It is defined only when the conditioning event has non‑zero probability, and it reflects how knowledge about one event updates the likelihood of another.

All probabilities are essentially conditional; there is no absolute probability. Context, models, and assumptions always influence probability statements, whether we discuss tomorrow’s rain on Earth, Mars, or elsewhere, or the bias of a coin.

Reference: Osvaldo Martin, “Python Bayesian Analysis”