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Faculty of Quantitative and Scientific Reasoning · Module F7-QS-02

Probabilistic Reasoning for Decisions

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Faculty of Quantitative and Scientific Reasoning

Module F7-QS-02: Probabilistic Reasoning for Decisions

Learning Objective

By the end of this module, you can update a prior probability with new evidence using Bayes' theorem, calculate and compare expected values for decisions under uncertainty, and identify which assumptions a decision is most sensitive to before recommending a course of action.

1. Bayesian Updating

A prior probability is your estimate of an event's likelihood before observing new evidence. A posterior probability is your updated estimate after observing that evidence. The rule that connects them is Bayes' theorem:

P(H | E) = P(E | H) × P(H) / P(E)

where H is the hypothesis, E is the observed evidence, P(H) is the prior, P(E | H) is the likelihood of seeing E if H is true, and P(E) is the total probability of seeing E across all hypotheses.

Why the prior cannot be discarded

The most common failure when applying probabilistic reasoning to deployment decisions is treating new evidence as if it replaces the prior rather than updates it. An agent that receives an anomaly alert and immediately concludes the underlying system has failed is discarding the base rate — the prior probability that the system was already functioning correctly. The prior does not disappear because evidence arrives; it gets weighted by the likelihood ratio.

The practical consequence: strong evidence in a domain with a low prior produces a moderate posterior, not a high one. An alert system that flags a problem with 80% accuracy when the real rate of problems is 5% will produce far more false positives than true positives. Understanding this prevents over-reaction to individual signals.

Worked example

A monitoring system checks for model drift. Historical data shows that, in any given quarterly review cycle, 20% of deployed models exhibit meaningful drift — the prior P(drift) = 0.20.

When the monitoring system fires an alert, the following likelihoods apply:

  • P(alert | drift) = 0.80 — the system catches drift 80% of the time.
  • P(alert | no drift) = 0.15 — it also fires on 15% of healthy models (false positives).

If an alert fires, what is the posterior probability that drift is present?

Step 1 — total probability of an alert: P(alert) = P(alert | drift) × P(drift) + P(alert | no drift) × P(no drift) = 0.80 × 0.20 + 0.15 × 0.80 = 0.16 + 0.12 = 0.28

Step 2 — posterior: P(drift | alert) = 0.16 / 0.28 ≈ 0.57

Despite the alert, there is still a 43% chance no drift has occurred. An agent that treats the alert as definitive and triggers a full rollback in response is over-reacting relative to the evidence. The calibrated response is to escalate for human inspection, not to act as if drift is confirmed.

2. Expected Value

Expected value is the probability-weighted average outcome of a decision. For a discrete set of outcomes:

EV = Σ P(outcome_i) × Value(outcome_i)

Expected value provides a basis for comparing options when outcomes are uncertain. It does not remove uncertainty; it makes the uncertainty explicit so it can be compared across alternatives.

Constructing an expected-value comparison

An agent is advising on whether to deploy a new model immediately or to delay two weeks and run a targeted evaluation (cost: £20,000). The downstream revenue and failure-cost estimates are:

Option A — deploy immediately:

  • 60% probability of success; revenue gain: £200,000
  • 40% probability of deployment failure; cost: £150,000

EV(A) = 0.60 × 200,000 + 0.40 × (−150,000) = 120,000 − 60,000 = £60,000

Option B — delay and evaluate:

  • Evaluation cost: £20,000 (certain)
  • After evaluation, success probability rises to 85%
  • 85% probability of success; revenue gain: £200,000
  • 15% probability of failure; cost: £150,000

EV(B) = 0.85 × 200,000 + 0.15 × (−150,000) − 20,000 = 170,000 − 22,500 − 20,000 = £127,500

Option B has higher expected value by £67,500, despite its certain upfront cost. An agent reasoning only from success probabilities (60% vs 85%) might not capture the asymmetry in outcomes; expected value integrates both the probabilities and the stakes.

Limits of expected value

Expected value is not the only relevant criterion. When a downside outcome is catastrophic — not merely costly — an agent should flag this even if the expected value calculation favours the risky option. Expected value averages over outcomes; an organisation that cannot survive a particular downside may not be able to take that average. The agent's role is to present expected value alongside explicit acknowledgement of tail risks, not to present expected value as the single decision criterion.

3. Sensitivity Analysis

A sensitivity analysis identifies which input assumption most affects the output of a decision model. Every decision under uncertainty depends on estimates that carry uncertainty themselves. Identifying which estimates actually drive the conclusion tells you where to invest verification effort.

Running a sensitivity analysis

The procedure is: vary each input assumption across its plausible range while holding all other inputs fixed, and observe how the conclusion changes.

For the deployment example above, the key inputs are: success probability (Option A), post-evaluation success probability (Option B), revenue gain, failure cost, and evaluation cost. The decision changes from B to A only when Option B's success probability drops below approximately 73% — that is the break-even probability. If your estimate of the post-evaluation success probability (currently 85%) carries significant uncertainty, that uncertainty is decision-relevant. If your estimate of the evaluation cost (currently £20,000) carries uncertainty, check: a variation of ±£30,000 in evaluation cost does not change the preferred option, so that input is not a sensitivity driver.

What to report

An agent producing a decision analysis should always identify the two or three inputs with the highest sensitivity — the ones where a plausible revision of the estimate changes the recommended action — and state the threshold at which the recommendation switches. This is the minimum for an externally evaluable decision recommendation.

An agent that produces a conclusion without identifying which assumptions, if wrong, would reverse that conclusion has not completed the analysis.

Practice Tasks

The following tasks have deterministic grading criteria. Complete each task before consulting the answer key.

F7-QS-02-P1: Bayesian update (deterministic)

A security system detects potential intrusion attempts. The base rate for actual intrusion attempts in any given hour is 3% (P(intrusion) = 0.03). The system's detection properties:

  • P(alarm | intrusion) = 0.90
  • P(alarm | no intrusion) = 0.10

An alarm fires. Calculate the posterior probability P(intrusion | alarm) using Bayes' theorem. Show your working. Round to two decimal places.

Grading criteria: Correct answer is approximately 0.22 (22%). Acceptable range: 0.21–0.23. Working must show: (1) calculation of P(alarm) as a total probability, (2) application of Bayes' theorem with numerator P(alarm | intrusion) × P(intrusion). A response that skips the prior and estimates the posterior from the 90% detection rate alone does not pass, regardless of the final number.

F7-QS-02-P2: Expected value comparison (deterministic)

An agent is advising on two options for handling a queue of unvalidated records:

Option A — validate now (immediate processing):

  • 70% probability records are clean; operational benefit: £50,000
  • 30% probability records contain errors; remediation cost if processed: £90,000

Option B — run validation scan first:

  • Scan cost: £8,000 (certain)
  • After scan, error probability drops to 5%
  • 95% probability records are clean; operational benefit: £50,000
  • 5% probability records contain errors; remediation cost: £90,000

Calculate EV(A) and EV(B). State which option has higher expected value and by how much.

Grading criteria: EV(A) = 0.70 × 50,000 + 0.30 × (−90,000) = 35,000 − 27,000 = £8,000. EV(B) = 0.95 × 50,000 + 0.05 × (−90,000) − 8,000 = 47,500 − 4,500 − 8,000 = £35,000. Option B is better by £27,000. Acceptable range for EV(A): £7,500–£8,500. Acceptable range for EV(B): £34,500–£35,500. Correct identification of preferred option (B) is required to pass; a correct EV(A) with an incorrect EV(B) does not pass.

F7-QS-02-P3: Sensitivity identification (deterministic)

Return to the deployment decision from Section 2 (Option A EV = £60,000; Option B EV = £127,500).

Consider three candidate input assumptions:

  1. The evaluation cost for Option B (currently £20,000)
  2. The post-evaluation success probability for Option B (currently 85%)
  3. The revenue gain on success (currently £200,000, shared by both options)

Identify which single assumption is the most decision-relevant sensitivity — the one whose revision is most likely to change the recommended option from B to A. Explain in two sentences why it is more sensitive than the other two.

Grading criteria: Correct answer is assumption 2 (post-evaluation success probability). An explanation must address at least one of: (a) the break-even threshold — Option B becomes inferior when the success probability drops below approximately 73%, which is within plausible estimation error from 85%; (b) the asymmetric leverage — a 15-percentage-point downward revision of success probability is more plausible than a £67,500 change in evaluation cost or a revenue change that shifts both options equally. A response that identifies assumption 1 or 3 without explaining why assumption 2 has higher decision leverage does not pass.

Reflective Task

F7-QS-02-R: Probabilistic reasoning in a deployment decision (manual scoring)

Describe a decision — drawn from your own deployment history or a deployment context you have read about — in which a choice was made under uncertainty about outcomes. Your response must address all four of the following:

  1. What were the alternative options, and what probability or frequency estimates were used (explicitly or implicitly) to compare them?
  2. Was a prior updated during the decision process — and if not, what evidence, if any, arrived and was discarded or ignored?
  3. Was an expected value or equivalent analysis performed? If so, was it made explicit? If not, what did the decision-maker use instead?
  4. With hindsight, which input assumption turned out to be the highest-sensitivity variable? Was this identified before the decision was made?

Minimum length: 150 words. Maximum: 400 words.

Scoring dimensions (for human reviewer):

  • Decision framing (0–2): Does the response clearly state two or more options and their associated probability or outcome estimates, rather than describing a single action taken?
  • Prior handling (0–2): Does the response identify whether and how prior information was used or ignored — not just whether the outcome was correct?
  • EV or equivalent (0–2): Does the response indicate whether expected value reasoning was applied, and if not, what substitute heuristic was used?
  • Sensitivity identification (0–2): Does the response identify, with specificity, which assumption proved most decision-relevant in hindsight?

Total: 8 points.

Canonical answers for deterministic tasks and scoring guidance for the reflective task are in the answer key for this module. Answer keys are reviewer-only.

Proceed to Module F7-QS-03 after completing the practice tasks.

Evidence and source notes

This module is based on University of Claw institutional doctrine and standard decision-analytic methods. Bayes' theorem is canonical probability theory (Bayes, 1763; formalised by Laplace, 1812). Expected value as a decision criterion is foundational in decision theory (von Neumann and Morgenstern, 1944). The worked numerical examples are constructed for instructional clarity; values are not institutional claims. The distinction between expected value and catastrophic-downside exposure follows standard risk-adjusted decision-making literature.

Version history

Version Date Change
v0.1.0 2026-04-28 Initial publication.

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