Expected value and decision trees, explained
Expected value is the average outcome, weighted by how likely each outcome is. In a decision tree it is the number on every node, and reading it back from the ends tells you which option is worth most. This guide uses one small example, a shipment you can insure or not, and every number on the page is computed from it.
Published 2026-09-27. By the DecisionBranch team.
What expected value is
Expected value is what an uncertain outcome is worth on average: each possible result multiplied by its probability, added up. It is the number you would get per go if you could take the same gamble many times.
It is not the most likely result and it is not a result you will necessarily ever see. A coin flip that pays 100 on heads and nothing on tails has an expected value of 50, and 50 never comes up. What it gives you is a single number that lets you compare options fairly, which is exactly what a decision needs.
The formula, and a one-line example
EV = Σ (probability × payoff). For the uninsured shipment: 96% × 40,000 + 4% × 0 = 38,400.
The shipment is worth 40,000 if it arrives and nothing if it is lost, and it is lost 4% of the time. Multiply each payoff by its probability and add: 96% × 40,000 + 4% × 0 = 38,400. On average, sending it uninsured is worth 38,400.
Insured, the shipment is worth 40,000 whether it arrives or not, because the insurer pays out. That is an expected value of 40,000, minus the 1,200 premium: 38,800. Already the answer is visible: insuring is worth 400 more on average.
Expected value in a decision tree
In a decision tree, every chance node carries the expected value of its branches, every decision node carries the value of its best branch, and the root carries the value of the whole decision. The lit path is the recommendation.
# A 40,000 shipment. Insurance costs 1,200 and pays the full value if it is lost.
Insure the shipment?
Insure (cost 1200)
Arrives fine (p 0.96): 40000
Lost or damaged (p 0.04): 40000 # the insurer pays out
Don't insure
Arrives fine (p 0.96): 40000
Lost or damaged (p 0.04): 0The square is the decision, the circles are the chance nodes, and the badge on each is its expected value. The cost of insuring sits on its branch, so the payoffs stay gross and the engine subtracts the premium once, on the way back to the root.
Rollback, step by step
Rollback is how the numbers get onto the tree: start at the ends, replace each chance node with its expected value, replace each decision node with its best branch after costs, and repeat until you reach the root.
- 1Insure is a chance node. Its value is the probability-weighted average of its branches: 96% × 40,000 + 4% × 40,000 = 40,000.
- 2Don't insure is a chance node. Its value is the probability-weighted average of its branches: 96% × 40,000 + 4% × 0 = 38,400.
- 3Insure the shipment? is a decision. You would pick the best branch: Insure = 38,800 (40,000 − 1,200 cost); Don't insure = 38,400. Best: Insure at 38,800.
- ✓So the whole decision is worth 38,800, and the recommendation is Insure.
For a bigger tree the steps get longer but never harder; it is the same two rules applied from the leaves inward. Paste any tree into the analysis calculator and it writes the steps out like this.
The switch point
The switch point is the value of an input at which the recommendation changes. Here, insuring stops being worth it if the chance of losing the shipment falls below 3%; it is 4% now.
You can find it by hand: not insuring is worth 40,000 × (1 − p), and insuring is worth 38,800 whatever p is. They are equal when 40,000 × (1 − p) = 38,800, so p = 1,200 ÷ 40,000 = 3%. Below that loss rate, take the risk; above it, buy the cover. The engine finds the same point by sweeping the probability and bisecting where the root choice flips.
The distance between your estimate and the switch point is how wrong you can be before it matters. That is the most useful number a decision tree produces, and it is why the tree is worth more than the picture.
What expected value does not tell you
Expected value hides the spread. Two options with the same average can have completely different worst cases. The risk profile lists every ending under an option and its probability, so you can see the chance of a bad outcome before you commit.
Insured, there is one ending: 38,800, always. Uninsured, there is a 4% chance of ending with nothing. The averages are 400 apart; the worst cases are 40,000 apart. For a company that ships every week, the average is what matters. For one shipment that would sink the business if it was lost, the worst case is.
Value of information
The expected value of perfect information is the most you should pay to know an uncertain outcome before you decide. Here, knowing in advance whether this shipment will be lost is worth 1,152.
With perfect information you would insure only the shipments that will be lost and save the premium on the rest: 96% × 40,000 + 4% × 38,800 = 39,952, against 38,800 without it. The difference, 1,152, is the ceiling on what a perfect forecast would be worth. A real forecast is imperfect and worth less, which is the calculation the oil lease example works through.
Questions
Short answers to the questions that come up most.
- Is expected value the same as expected monetary value (EMV)?
- Yes. EMV is expected value when the payoffs are money. The arithmetic is identical for any unit.
- What if the probabilities are guesses?
- Then sweep them. The switch point tells you how far a guess can be wrong before the recommendation changes. If it is far, the guess does not matter; if it is close, that is the number to firm up.
- Should I always pick the highest expected value?
- On average, yes. For a one-off decision with a large downside, also look at the risk profile: a slightly lower expected value with no chance of ruin can be the better choice.
- Where do costs go?
- On the branch you pay them on, in brackets: (cost 1200). Payoffs stay gross. The rollback subtracts each cost on the path, so a cost is never counted twice.