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.
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.
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.
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.
6. Distributions, risk tails and rare events
A mean value does not describe mission risk by itself. Two distributions can have the same mean while assigning very different probability to a dangerous region. For a Gaussian variable, standard deviation describes spread around the mean; for failures or extreme environmental events, asymmetric or heavy-tailed models may be more appropriate. Engineers therefore ask which part of the distribution drives the decision: nominal behavior, the 95th percentile, a credible worst case or the probability of crossing a defined threshold.
7. Updating probability when new evidence arrives
Mars decisions are sequential. Before a test, a team has an estimate of reliability; after each result, that estimate should change. Bayesian reasoning formalizes the process: a prior distribution is combined with the likelihood of new evidence to produce a posterior distribution. The method guards against two opposite mistakes, treating one successful test as proof of perfection or ignoring a large body of test evidence because an early model looked convincing.
8. Useful Monte Carlo: simulate a decision, not merely a cloud of numbers
Monte Carlo analysis draws many combinations of uncertain parameters from defined distributions. It becomes useful when several nonlinear uncertainties interact, such as mass, efficiency, duration, temperature, availability and consumption. The result is only as credible as the input distributions and correlations. One hundred thousand runs based on unjustified assumptions create a very precise error. A useful report therefore shows assumptions, sensitivity and the fraction of cases that violate a mission criterion.
Decision case: a comfortable mean can hide a dangerous tail
Assume a lander has comfortable mean propellant margin but a small fraction of simulations ends close to minimum reserve. The mean looks reassuring. Engineers therefore inspect percentiles and tail cases: if one percent of trajectories consume nearly all reserve, the decision depends on the consequence of that one percent, the credibility of the input distributions and whether an operational diversion rule can reduce exposure. A distribution is not decoration; it should lead to a design or operational decision.
9. Worked example step by step
A critical chain contains 8 independent series elements, each with a 0.995 probability of success during the relevant phase. The probability that all eight succeed is 0.995^8 ≈ 0.9607, or about 96.1%. The probability of at least one failure is therefore 1 − 0.9607 = 0.0393, or 3.93%. If the system requirement is at least 99% success, slightly improving every component may not be enough; the design may need functional redundancy or fault tolerance. The calculation shows why long series chains rapidly erode system reliability.
10. Progressive exercise
A subsystem contains five series functions at 0.998 reliability each and one redundant function made of two independent channels at 0.97. Calculate the redundant-function reliability and then the full-chain reliability. Repeat conceptually when a common-cause mechanism has a 2% probability of disabling both redundant channels together.
11. Reasoned solution
Two independent 0.97 branches succeed with 1 − (1 − 0.97)² = 0.9991. Five 0.998 series functions give 0.998⁵ ≈ 0.9900. The combined chain is about 0.9891 before common cause is included. A two-percent common-cause event means the redundancy cannot be credited as perfectly independent, so real reliability falls further.
12. Validation mini-project
Build a probabilistic model of one Mars operations day with at least five uncertain events, two correlations and an explicit failure criterion. Compare nominal, 95th-percentile and Monte Carlo results, then identify which uncertainty deserves the next test campaign.