Expected Value and Risk for the FE Civil Exam

NCEES FE Civil spec: Engineering Economics, D. Uncertainty (e.g., expected value and risk). FE Reference Handbook: Engineering Probability and Statistics chapter (expected value) and Engineering Economics chapter.

Real projects face uncertain floods, prices and demand. The FE Civil exam handles this with expected value: weight each outcome by its probability and compare options on their expected cost or worth. Some questions add a measure of risk, usually the standard deviation of the outcomes.

What the NCEES spec covers

Section D lists uncertainty, including expected value and risk. Typical problems: choose a cofferdam height or culvert size given flood probabilities, find the expected cost of a repair, or compare two investments with the same expected value but different spreads.

Key concepts

Probabilities must sum to 1 across all outcomes of an option. If a problem gives only some, the remainder is the "nothing happens" case.
Expected value is a long-run average. A single project will not land on it, but over many similar decisions it minimizes total cost.
Risk is the spread of outcomes, measured by variance or standard deviation. With equal expected values, a risk-averse owner prefers the smaller spread.
Annual probability. A flood with a return period T years has a probability 1/T of being exceeded in any one year, and $1 - (1 - 1/ T)^{n}$ of being exceeded at least once in n years.
Time value still applies. Expected annual damages can be converted to PW with (P/A) like any other annual cost.

Core formulas

E [X] = i \sum p_{i} x_{i} i \sum p_{i} = 1

σ^{2} = i \sum p_{i} (x_{i} - E [X])^{2} σ = σ^{2}

Expected total cost = first cost + \sum (probability \times loss)

Where it is in the FE Reference Handbook

Expected value, variance and standard deviation of a discrete distribution are defined in the Engineering Probability and Statistics chapter. The Engineering Economics chapter provides the interest factors if expected annual costs must be converted to present worth.

Worked example

A contractor must choose a cofferdam for one construction season. A low cofferdam costs $50,000 and has a 20% chance of being overtopped. A high cofferdam costs $90,000 and has a 5% chance. Overtopping causes $300,000 of damage either way. Which option has the lower expected cost, and which is less risky?

E_{low} = 50, 000 + 0.20 (300, 000) = $110, 000

E_{high} = 90, 000 + 0.05 (300, 000) = $105, 000

The high cofferdam has the lower expected cost. For risk, the low option's outcomes are $50,000 (p = 0.80) or $350,000 (p = 0.20):

σ_{low} = 0.80 (50, 000 - 110, 000)^{2} + 0.20 (350, 000 - 110, 000)^{2} = $120, 000

The high option's outcomes are $90,000 (p = 0.95) or $390,000 (p = 0.05):

σ_{high} = 0.95 (90, 000 - 105, 000)^{2} + 0.05 (390, 000 - 105, 000)^{2} = $65, 400

The high cofferdam wins on both expected cost and risk.

Common mistakes

Forgetting to add the first cost, which is certain, to the expected damage.
Using probabilities that do not sum to 1 because the "no loss" outcome was left out.
Choosing on expected value alone when the question asks about risk.
Treating a 100-year flood as one that happens once every 100 years. It has a 1% chance every year.

See the expected value example, and the statistics lesson for distributions.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

A repair has a 20% chance of costing $50,000 and an 80% chance of costing $5,000. What is the expected cost? See the worked solution or practice the Engineering Economics deck
A project has a 20% chance of a $100,000 gain and an 80% chance of a $10,000 loss. What is its expected net value? See the worked solution or practice the Math & Statistics Refresher deck

Related Calculators and Examples

Frequently Asked Questions

How do you calculate expected value?

Multiply each possible outcome by its probability and add the results. Check first that the probabilities of all outcomes sum to 1.

How is risk measured in engineering economics?

Usually by the variance or standard deviation of the outcomes. Two options with the same expected value can carry very different risk.

What is the chance of a 100-year flood during a 10-year project?

The annual exceedance probability is 0.01, so the chance of at least one exceedance in 10 years is 1 - 0.99^10, which is about 9.6 percent.

Back to all Engineering Economics topics