Lecture Seven · 9 June 2026

07Lecture Seven

Risk & Decision
under Uncertainty

When the inputs are not numbers but distributions — decision trees, expected value, utility, and Monte Carlo.

Instructor

Dr. Zhijiang Chen

Session

No. 07 of 16

Date

9 June 2026

Room

YF108

Duration

110 minutes

Format

Lecture + Simulation

Today: probabilities meet PV. Bring Python or a spreadsheet.

Lecture VIIAgenda02 / 20

Today's Plan

Probability for engineers.

§	Topic	Minutes
I.	Risk vs uncertainty — terms first	10
II.	Decision trees & expected monetary value	25
III.	Utility & risk attitude	15
IV.	Monte Carlo simulation	25
—	Discussion: risk-attitude self-assessment	10
V.	Software-specific risks & bias	15
	HW7, questions	10

Part — One

Risk vs uncertainty —
a vocabulary the textbooks
insist on.

§ IVocabulary04 / 20

Knight, 1921

The classical distinction.

Risk

Outcomes uncertain, but probabilities known. Roll of a fair die. Software example: a deployment with a known failure rate.

Uncertainty (true)

Outcomes uncertain AND probabilities unknown. New AI feature adoption. We act as if probabilities exist, but they're really subjective estimates.

Most software-economics problems are technically uncertainty masquerading as risk. We assign probabilities anyway — the act of estimation is half the value.

Part — Two

Decision trees —
think out loud, on paper.

§ IITree structure06 / 20

Decisions, chances, terminals

A decision tree is three node types.

□ Decision node

Branches the engineer controls. Pick the branch with the highest expected value.

○ Chance node

Branches nature controls. Multiply each branch's value by its probability; sum.

△ Terminal

An end state. Record the payoff (or cost).

Build the tree left-to-right (in time order). Roll back right-to-left, computing expected values at chance nodes and choosing maxima at decision nodes.

§ IIWorked example07 / 20

Should we ship the AI feature now?

EMV on a real decision.

Two options: A: ship now ($60K dev cost). B: spend $40K on a 6-week customer-discovery study, then decide.

Branch	P	Outcome NPV	Contribution
A: ship · high demand	0.4	+$300K	+$120K
A: ship · low demand	0.6	−$60K	−$36K
EMV(A)			+$84K
B: study · then ship if signal	—	—	+$108K *

* The study clarifies the probability, so the conditional ship decision is informed. Despite the $40K cost, the expected outcome rises.

The study creates option value — the right to ship only when conditions favour it. Information has measurable economic worth.

Part — Three

III

Utility & risk attitude —
not everyone treats
a 50/50 the same.

§ IIIUtility functions09 / 20

Beyond expected value

Risk attitude shapes the decision.

EMV treats $1M with certainty as equal to a 50/50 shot at $2M. In practice, decision-makers differ:

Attitude	Utility shape	Will accept
Risk-averse	Concave (log, square root)	Smaller, more certain payoffs
Risk-neutral	Linear	Anything with EMV ≥ 0
Risk-seeking	Convex (exponential)	Lottery-style large variance bets

Most software organisations are moderately risk-averse: a $100K loss hurts more than a $100K gain helps. Frame your decision tree with that in mind.

When you present a positive-EMV project that has substantial downside, expect resistance — and prepare a hedging argument.

Part — Four

Monte Carlo simulation —
let the inputs roll.

§ IVThe technique11 / 20

Probability distributions in, NPV distribution out

Monte Carlo in 4 steps.

Identify the inputs. For each, choose a probability distribution (uniform, normal, triangular, lognormal).
Draw N random samples (typically N = 10,000+) from each input distribution.
For each sample tuple, compute the NPV with your normal formula.
Histogram the resulting N NPVs. Report P10, P50, P90, and the probability of NPV < 0.

P(NPV > 0) is the modern, intelligible answer to "should we do this?"

§ IVWorked output12 / 20

A 10,000-trial result

Reading a Monte Carlo result.

Statistic	Value	Interpretation
Mean NPV	+$112K	Average across 10K worlds.
P10 NPV	−$45K	10% chance of NPV being below this.
P90 NPV	+$310K	10% chance of NPV exceeding this.
P(NPV > 0)	76%	Three-in-four chance of value creation.

"Mean NPV is $112K with a 24% chance of loss" communicates more than "NPV is $112K." It is also harder to misinterpret.

Discussion10 minutes13 / 20

Find your risk attitude

Would you take a 50/50 shot at $1M or a guaranteed $400K?

Vote first. Then in pairs, discuss: why? What does that reveal about your personal utility curve?

Now replay: would your answer change if the gamble was a 50/50 at $10M vs $4M guaranteed? Why does the ratio matter so much less?
Engineering management often demands risk-averse capital allocation. As an individual contributor, are you more or less risk-tolerant than your manager? Why might that be?

Part — Five

Software-specific risks —
the ones your tree
should always have.

§ VRisk catalogue15 / 20

Seven categories

A starter taxonomy for software-project risk.

Category	Example	Mitigation
Schedule	Slip past launch window	Buffer; scope discipline
Scope	Requirements creep	Change control; MVP first
Technical	Approach proves infeasible	Spike / prototype early
Personnel	Key engineer departs	Knowledge sharing; redundancy
Vendor	SaaS price doubles, vendor pivots	Multi-vendor; escape clauses
Model drift	LLM behaviour changes mid-contract	Pinned versions; eval harness
Regulatory	New AI law forces retrofit	Compliance monitoring; design margin

"Model drift" and "regulatory" are the categories most teams underweight in 2026. A model that performs at 92% accuracy today may quietly drift to 86% next quarter.

§ VEstimation bias16 / 20

Why our subjective probabilities are usually wrong

Three biases to budget against.

Optimism

P10 outcomes routinely come true 25–35% of the time, not 10%. Our pessimistic case is rarely pessimistic enough.

Anchoring

The first number on the table dominates the rest of the conversation. Estimate independently before sharing.

Planning fallacy

We systematically underestimate time and effort, even for tasks we've done before.

Adjust subjectively for bias. A useful rule of thumb: take your "realistic" estimate, then ask "what would I think if this had to ship in half the time?" Use that as your new pessimistic.

BridgeTo Lecture 817 / 20

Tomorrow: how big is this project, exactly?

Lectures 8 & 9 turn the right side of our cash-flow models — the costs — into something we can estimate even at proposal stage. Function Points first, COCOMO II second.

Group project briefs are released after Lecture 8. Teams should be forming now.

HomeworkDue Lecture 918 / 20

Homework 07 — due Thursday 11 June

Add probability to your NPV.

Build a decision tree for a real choice your team is facing (or has faced).
Run a 10,000-trial Monte Carlo on the same project. Report mean, P10, P50, P90, P(NPV > 0). Plot the histogram.
One paragraph: which uncertainty would you spend $5K of research budget to reduce — and why?

RecapWhat to remember19 / 20

What today bought you.

A decision tree forces you to make assumptions visible. That alone is usually worth the exercise.
Monte Carlo turns a single NPV into a distribution. "P(NPV > 0) = 76%" is more honest than a single number.
Our subjective probabilities are biased. Build counter-bias into your scenarios; the P10 should hurt.

EndLecture Seven20 / 20

Questions & conversation.

Dr. Zhijiang Chen
Software Engineering Economics · Summer 2026
frostburg-state-university.github.io/bju

Risk & Decisionunder Uncertainty

Probability for engineers.

Risk vs uncertainty —a vocabulary the textbooksinsist on.

The classical distinction.

Risk

Uncertainty (true)

Decision trees —think out loud, on paper.

A decision tree is three node types.

□ Decision node

○ Chance node

△ Terminal

EMV on a real decision.

Utility & risk attitude —not everyone treatsa 50/50 the same.

Risk attitude shapes the decision.

Monte Carlo simulation —let the inputs roll.

Monte Carlo in 4 steps.

Reading a Monte Carlo result.

Would you take a 50/50 shot at $1M or a guaranteed $400K?

Software-specific risks —the ones your treeshould always have.

A starter taxonomy for software-project risk.

Three biases to budget against.

Optimism

Anchoring

Planning fallacy

Tomorrow: how big is this project, exactly?

Add probability to your NPV.

What today bought you.

Questions & conversation.

Risk & Decision
under Uncertainty

Risk vs uncertainty —
a vocabulary the textbooks
insist on.

Decision trees —
think out loud, on paper.

Utility & risk attitude —
not everyone treats
a 50/50 the same.

Monte Carlo simulation —
let the inputs roll.

Software-specific risks —
the ones your tree
should always have.