The Expected Value Habit

Why this matters

Most owner decisions are bets: bid the big job or not, pay for the booth or not, fight the warranty denial or eat it. Owners tend to decide these on the best case ("if we win this, we are set for the summer") or the worst case ("what if it goes sideways"). Both ignore how likely each case is. Expected value is the habit of weighing every outcome by its chance of happening and adding them up, and it takes a napkin, not a spreadsheet. The payoff is not precision. It is that you stop taking bets that only look good because of how good the win would feel, and you notice the cheap lever that turns a marginal bet into a good one.

The idea in one line

For each option, list what could happen, put a rough probability on each outcome, value each outcome in one unit, multiply each value by its probability and add. The option with the higher total is the better bet on average. Then check one more thing: whether you can survive the bad outcome if it comes.

The unit, and why one unit matters

You cannot add hours of the owner's time to a fraction of a job's margin to a lost customer unless they are in the same unit. This card uses margin hours: the gross margin the shop earns from one average billed labor hour. A booth fee "worth about 35 margin hours" means it costs as much as the margin on 35 typical billed hours. A tech pulled off billable work for a day costs the margin hours he would have billed.

Effort hours that would not otherwise have been billed (office time, the owner's evening) are a different currency. Keep them in their own line rather than pretending they are margin hours. Mixing the two is the most common way an expected value comes out flattering.

The procedure

1. Name every option, including doing nothing. "Bid or do not bid" is two options; "bid at our normal price, bid high, or pass" is three. Leaving out the do-nothing option is how owners end up comparing two ways of doing something that should not be done at all.

2. List the outcomes for each option. Usually two or three are enough: win and lose; win clean, win with an overrun, and lose. More branches rarely change the answer and always slow you down.

3. Put a rough probability on each outcome, from your records where you can. Use your bid log, your close rate by lead source, your history with that manufacturer. Where you have no record, a peer's experience or an honest guess stated as a range is fine. The starting number should be how often things like this have gone your way before. See related: Base Rates: The Number You Should Start From.

4. Value each outcome in the one unit, including what the option displaces. If winning the big job means the crew stops doing its normal residential work for six weeks, the value of winning is the job's margin minus the margin that crew would have earned anyway. Opportunity cost is its own card; here it is just a line you must not leave out. See related: Opportunity Cost: The Job You Did Not Take.

5. Multiply and add. Probability times value for each outcome, then sum, then subtract any cost you pay whatever happens (bid prep, a booth fee).

6. Check the bad branch. Expected value describes the average over many bets. You only get to make this one once. If the bad outcome would sink the shop, a positive expected value does not rescue the bet.

7. Look for the lever. Before deciding, ask which single input moves the total most. It is usually one cost line or one probability, and it is often cheap to change.

Application one: bidding the big job

A 14-person roofing and gutter shop is invited to bid a six-week commercial reroof that would take its three-person replacement crew off residential work for the duration.

• Win probability: the shop's commercial bid log shows about 1 in 3 won.
• Value if won: the job's margin is about 420 margin hours. The crew's normal residential output over six weeks in peak season is about 330 margin hours. So winning nets 420 minus 330, which is 90 margin hours.
• Overrun risk: on past commercial jobs, about 1 in 5 ran into a problem (deck rot, a late material delivery) that cost about 120 margin hours. If that happens, the net is 90 minus 120, a loss of 30.
• Bid prep: about 20 margin hours of the owner's time pulled off selling residential work.

The arithmetic:

• if won: 0.8 times 90 plus 0.2 times minus 30, which is 72 minus 6, so 66 margin hours
• times the 1 in 3 chance of winning: 22 margin hours
• minus 20 for the bid prep, paid win or lose: about 2 margin hours

Barely positive. The excitement of a big commercial contract was carrying most of the appeal.

The lever: what the crew would otherwise be doing. The same bid in the slow season, when residential work over those six weeks would bring only about 80 margin hours, looks very different. Winning nets 420 minus 80, which is 340; with an overrun, 340 minus 120, which is 220. If won: 0.8 times 340 plus 0.2 times 220, which is 272 plus 44, so 316. Times 1 in 3 is about 105, minus 20 for the bid: about 85 margin hours. The same job is a weak bet in June and a strong one in November.

The bad branch: the worst case is the overrun, a net loss of about 30 margin hours plus the bid prep, around 50 in total. Unpleasant, survivable. So the owner bids in peak season only if the customer will accept a start date in the shoulder season, and otherwise passes.

Application two: the lead source

A nine-person HVAC shop is deciding whether to take a booth at the spring home show again. The last two shows produced 68 leads and 10 sold jobs between them, about 15 percent. Show leads tend to be replacements, so a sold job averages about 16 margin hours.

HOME SHOW BOOTH, spring          (unit: margin hours)

  Option 1: two techs staff the booth
    booth fee ......................... -35
    2 techs x 3 days x 7 billed hrs ... -42
    leads expected ........ 35
    close rate, last 2 shows: 10 of 68, about 15%
    sold jobs ............. about 5
    5 jobs x 16 margin hours .......... +80
    expected value .................... + 3

  Option 2: owner and office manager staff it
    booth fee ......................... -35
    owner off sales calls ............. -10
    same leads, same sold jobs ........ +80
    expected value .................... +35

Two assumptions sit under option 2, and both are stated so they can be checked. The office manager's booth time is effort she would not have billed anyway, so it is not counted as margin hours; it goes in its own line as about 24 hours of office attention the shop has to find. And the close rate holds because show leads are booked for an in-home estimate, where the sale is actually made, rather than closed at the booth.

The lever was the staffing line. Pulling two techs off billable work in spring cost more margin than anything else in the bet. Option 1 is a coin flip around zero; option 2 is clearly worth doing, if the office can absorb those 24 hours. Nothing about the show itself changed.

Application three: the warranty fight

A manufacturer has denied a warranty claim on a failed compressor because the unit was registered late. The part costs the shop roughly the margin on 25 billed hours. The options are fight it, eat it, or split it with the customer.

The shop's history: of 9 denials it has appealed, 4 were reversed, about 45 percent.

Fighting costs: about 2 hours of the tech's time pulling photos and service records (2 margin hours, because he would otherwise be billing), plus about 6 hours of office time drafting the appeal and chasing the rep, kept in its own line.

Expected value of fighting: 0.45 times 25 is about 11 margin hours, minus 2, so about 9 margin hours, for 6 office hours. Worth doing if the office has the time.

What changes the answer: the type of denial. When the owner splits the history by reason, denials for late registration specifically were appealed 4 times and reversed once, 1 in 4. On that rate: 0.25 times 25 is about 6, minus 2, so about 4 margin hours for the same 6 office hours. Marginal. The habit's real contribution here is not the appeal decision at all; it is noticing that registering every unit at install, a few minutes on the ticket, removes this kind of denial altogether.

What the habit does not do

It does not replace judgment about the bad branch. A bet with a positive average that carries a 1 in 10 chance of losing a key commercial customer, or of a cash crunch the shop cannot ride out, can still be wrong. Write the worst outcome in plain words next to the total and ask whether you could take it.

It does not make the probabilities right. A napkin calculation with an invented 80 percent win rate is worse than no calculation, because now the bad guess looks like analysis. Pull the number from your records wherever one exists, and where it does not, run the total at a pessimistic and an optimistic guess. If the sign stays the same across both, the decision is robust. If it flips, the decision depends on a number you do not know, and finding that number is the next move.

It does not price things that are not in the unit. Reputation, a relationship with a general contractor, a tech's morale after a brutal job: name them in a line under the total and decide whether they are large enough to override it. Usually they are not. Occasionally they are the whole decision.

How to verify you are doing it right

Write the expected value on the decision record when you make the bet, and look back after twenty or so of them. If the bets you rated positive have, as a group, come out ahead, the habit is working. If your positive bets keep losing, the probabilities are too optimistic, and your records will show which kind: win rates, close rates or overrun chances. See related: Keeping a Decision Journal.

References

• von Neumann, J., and Morgenstern, O., Theory of Games and Economic Behavior (1944), the formal basis of expected value in decisions
• See related: Base Rates: The Number You Should Start From; Opportunity Cost: The Job You Did Not Take; Keeping a Decision Journal; Stress-Testing a Plan With Three Scenarios