Probability, statistics and uncertainty propagation
This course develops a capability that was still missing from the core curriculum. It starts from concepts and units, builds the necessary calculations, then connects each method to real Mars engineering decisions.
Mastery objectives
- explain quantities, units and assumptions
- repeat at least one calculation by hand
- identify uncertainty, limits and failure modes
- turn the result into a decision for a Mars architecture
1. Missions use distributions, not single magic numbers
Mass, thrust, atmospheric density, navigation error and lifetime all have dispersion. A robust design asks not only for the expected value but for the range that remains compatible with mission success.
False precision is dangerous: uncertainty describes what is known, not how many digits a screen can display.
Engineering reflex. Identify what is measured, assumed and calculated, then state what would change the decision.
2. Mean, variance and standard deviation
For measurements xᵢ, the arithmetic mean is x̄ = Σxᵢ/n. Variance measures spread and the standard deviation σ has the same unit as the quantity.
Five flow measurements near 10 g/s should be summarised with their spread rather than by choosing the largest value as representative.
Engineering reflex. Identify what is measured, assumed and calculated, then state what would change the decision.
3. Propagating uncertainty
When an output depends on several inputs, final uncertainty depends on sensitivity to each variable. Independent errors can often be combined approximately, but correlation changes the result.
Two sensors with the same calibration bias are not two independent confirmations.
Engineering reflex. Identify what is measured, assumed and calculated, then state what would change the decision.
4. Monte Carlo simulation
Monte Carlo draws many input combinations from their distributions and recalculates the system each time. It does not remove uncertainty; it exposes the resulting distribution.
Thousands of trials can estimate how often thermal, propellant or landing constraints are violated.
Engineering reflex. Identify what is measured, assumed and calculated, then state what would change the decision.
5. Conditional probability and diagnosis
Evidence changes the probability of competing hypotheses. A common alarm may be weak evidence; a second independent indicator can change the diagnosis sharply.
Bayesian reasoning helps rank hypotheses during distant anomalies, but it never replaces testing and physical understanding.
Engineering reflex. Identify what is measured, assumed and calculated, then state what would change the decision.
Worked example step by step
Build a nominal case and a degraded variant. Write every input with units, convert to one coherent system, perform the calculation, then translate the result into a sentence. Finally vary the most uncertain parameter by ±20% and check whether the decision changes.
Progressive exercise
- Choose a Mars subsystem and list five inputs.
- Classify each input: measured, sourced, assumed or calculated.
- Calculate the nominal case.
- Inject uncertainty or a failure.
- Decide: continue, degrade, stop or reconfigure.
Reasoned solution
A good solution shows units, reasoning, sensitivity and the decision. A numerical result without physical interpretation is not a complete solution.
Validation mini-project
Produce a three-to-five-page engineering note with need, assumptions, diagram, calculation, uncertainty, injected failure, decision criterion and at least three primary sources.