Expected Value Explained: Formula and Practical Examples

A $15,000 freelance project isn’t automatically worth more than a $5,000 project, and expected value is the calculation that shows why. At a 30% chance of closing, the larger project’s expected revenue is $4,500. At a 90% chance, the smaller project’s expected revenue is also $4,500.

That’s expected value: multiply the value of each possible outcome by its probability, then add the results. Here, assuming a rejected proposal earns nothing, the two projects tie before costs.

I wouldn’t choose between them on that number alone. Proposal time, delivery costs, payment timing and the consequences of earning nothing could change my choice.

One article can’t settle every decision, so this one stays with the parts you can reuse: the formula, one complete worked example, the break-even probability a decision has to clear and the checks that still matter when the number comes out positive. In addition to that, I have put together an Excel worksheet you can download and fill in for your own decision, and illustrated study notes with a practice set and full solutions. I’ll point you to both as we go. Let’s get started.

What Is Expected Value?

Expected value is the starting point for almost every calculation about an uncertain outcome, from an insurance premium to a product launch. It is the average result of a decision, with every possible outcome counted in proportion to how likely it is, which textbooks call the probability-weighted average. If you faced the same decision many times over, expected value is what you’d get per attempt, on average.

For outcomes x₁, x₂, … x_n with probabilities p₁, p₂, … p_n, the expected value formula is:

$$E[X] = \sum_{i=1}^{n} p_i \, x_i = p_1 x_1 + p_2 x_2 + \dots + p_n x_n$$

The probabilities must add up to 1, or 100%. The outcomes can be money, hours, units sold, customers kept, anything you can put a number on.

The simplest bet shows how it works. A coin flip pays you $100 on heads and costs you $50 on tails.

$$E = (0.5 \times \$100) + (0.5 \times -\$50) = \$50 – \$25 = \$25$$

The expected value is $25 per flip. That number deserves a second look, because no single flip ever pays $25. Each flip pays $100 or takes $50, and $25 is what the two results average out to once they’re weighted by their chances. Expected value is an average, and it is not necessarily the most likely result or even a result that can occur in one attempt.

So the coin game has positive expected value, and it is still a game to play only if losing $50 on the next flip is money you can spare. A lower expected value can be the rational choice when a loss would hurt you more than the matching gain would help. That distinction runs through the rest of this article, and I’d rather you met it here than in the fine print.

How to Calculate Expected Value

The calculation itself is multiplication and addition. The work is in setting it up properly, because a setup mistake is where most wrong answers come from. This is the order that works.

1. Define the decision and the period. Name the choice you’re pricing and the window it covers, whether that is one year, 3 years or the life of the project.
2. List the scenarios. Write down every way this could turn out, from a big win to a serious loss.
3. Put a net outcome on each scenario. Net means after the costs that belong to that scenario.
4. Estimate the probabilities. Use what you have: your own history with similar decisions, industry base rates or an honest expert judgment.

Then multiply each outcome by its probability and add the results. Say you’re pricing a product launch, and the net results over the launch period look like this:

Outcome	Probability	Net outcome	Contribution to expected value
Massive success	10%	$500,000	$50,000
Solid success	25%	$200,000	$50,000
Modest success	30%	$50,000	$15,000
Break-even	15%	$0	$0
Minor loss	15%	-$75,000	-$11,250
Major failure	5%	-$300,000	-$15,000
Total	100%		$88,750

The expected value of the launch is $88,750. Read the rest of the table before you read that as good news. The same model gives a 20% chance of losing money and a worst listed outcome of losing $300,000, and these two numbers belong next to the expected value, not several sections later.

Before trusting any table like this, run a short setup check:

• the scenarios don’t overlap and nothing likely is missing, so the probabilities account for the whole model
• every outcome uses the same units and the same period
• the probability column adds up to exactly 100%

How to Use Expected Value to Compare Options

An expected value on its own tells you very little. It becomes useful next to the realistic alternative, which is usually doing nothing, or spending the same money and time somewhere else.

For the launch above, doing nothing is worth $0, so the $88,750 is the whole premium for taking the risk. If the same budget could earn a near-certain $50,000 elsewhere, the launch is worth $38,750 more in expectation than the safe route, and that $38,750 is what you’re being paid to accept a 20% chance of a loss. Whether that is a fair trade depends on how the loss would land on you, and the arithmetic can’t answer that part.

The same scale also works across options that look nothing alike. Say you have $10,000 for marketing and 3 ways to spend it:

Option	Chance it works	Revenue if it works	Revenue if it doesn’t	Expected revenue	After the $10,000 cost
Facebook ads	50%	$35,000	$5,000	$20,000	$10,000
Influencer campaign	25%	$100,000	$2,000	$26,500	$16,500
Trade show	90%	$20,000	$12,000	$19,200	$9,200

These figures are expected revenue less the campaign cost rather than profit, because product and delivery costs aren’t in the model. On that scale the influencer campaign leads, and it also fails 3 times in 4, leaving $2,000 of revenue against $10,000 spent. The trade show comes last on expectation and almost never disappoints. If the $10,000 is money you could lose without changing anything else in the business, the influencer campaign is the better bet. If losing it would mean cutting something that matters, the trade show is the better decision, and the lower number is the right answer.

How Likely Does Success Need to Be?

This is the question I’d add to every expected value calculation, because it turns a guessed probability into a threshold you can test. Instead of defending one estimate, work out the probability at which the decision breaks even, then ask whether your evidence clears it.

Take a software feature. Building it costs $80,000. If it succeeds and brings in new customers, it’s worth $300,000 in net benefit over 3 years. If it fails, it still keeps a few customers around and is worth $20,000. Both benefits are counted before the build cost and without discounting, which keeps the example simple. Say your best estimate of success is 45%.

$$E = (0.45 \times \$300{,}000) + (0.55 \times \$20{,}000) – \$80{,}000 = \$66{,}000$$

That is positive, under those assumptions. The 45% is the weakest number in the whole calculation, so instead of trusting it, write the expected value as a function of the success probability p:

$$E(p) = p\,(\$300{,}000) + (1-p)\,(\$20{,}000) – \$80{,}000 = \$280{,}000\,p – \$60{,}000$$

Set it to zero and solve:

$$p = \frac{60{,}000}{280{,}000} \approx 21.4\%$$

The feature needs better than a 21.4% chance of success to have positive expected value, measured against doing nothing. That changes the question. You no longer need to believe that 45% is an exact forecast. You need evidence that success is more likely than about 1 in 5, and you need to know whether the decision still works under a less optimistic estimate.

Assumed success probability	Expected net value
15%	-$18,000
20%	-$4,000
21.4%	$0 (break-even)
25%	$10,000
30%	$24,000
45%	$66,000

That threshold comes with two cautions. It is measured against doing nothing, so a valuable alternative use of the same $80,000 raises the bar. And it is only a calculation. It tells you what has to be true for the feature to make sense, and finding out whether it is true is still your job.

When Better Information Isn’t Worth Buying

The natural next move is to buy better information, say a $5,000 customer survey before building. Suppose the survey is right 80% of the time in both directions: 80% of features that test well succeed, and 80% of features that test poorly fail. Whether it is worth $5,000 depends on the whole strategy around it, not on the good result alone.

For a 45% base rate and an 80% accurate survey to hold together, a positive result has to come up 5 times in 12, about 42% of the time:

$$0.45 = 0.80\,q + 0.20\,(1-q) \quad \Rightarrow \quad q = \tfrac{5}{12}$$

With the survey result in hand, these are the moves on the table:

Decision after the survey	Expected feature payoff, before the survey fee
Build after a positive result	0.8 × $300,000 + 0.2 × $20,000 − $80,000 = $164,000
Build after a negative result	0.2 × $300,000 + 0.8 × $20,000 − $80,000 = −$4,000
Skip after a negative result	$0

So you build only after a positive result, and the survey route is worth:

$$\tfrac{5}{12}\,(\$164{,}000) – \$5{,}000 \approx \$63{,}333$$

Building without the survey was worth $66,000. Before its fee, the survey route is worth about $68,300, so the information adds about $2,300 in expected payoff, and it costs $5,000. Under these assumptions the survey lowers the expected payoff by about $2,700, even though it is exactly as accurate as promised. Better information is worth buying only when its expected improvement to the decision is bigger than its price, and you find that out by pricing the complete strategy rather than the favorable branch on its own.

When Positive Expected Value Isn’t Enough

A positive expected value earns an option a closer look and nothing more. There are 4 situations where the number on its own gives a wrong answer.

A Loss You Can’t Absorb

Two options can share an expected value and carry completely different risk. Option A pays $100,000 for certain. Option B pays $300,000 half the time and loses $100,000 the other half. Both have an expected value of $100,000, and only one of them can put you out of business.

The average also hides the rare disaster. A decision with a 99% chance of gaining $200,000 and a 1% chance of losing $10 million has an expected value of $98,000, which looks fine on paper. If the $10 million loss would end the business, the average is meaningless, because you won’t be around to collect the long-run result. Expected value describes a game you get to play many times. A one-time decision with a ruinous branch needs a different question first: can you survive the worst listed outcome? Sizing repeated bets so that you can, including the Kelly criterion, is its own topic, and the guide to the mathematics of risk assessment on this site covers it.

Inputs You Can’t Trust

An expected value can’t be more reliable than the probabilities that went into it, and a guessed probability is far more useful written down with its reasons than defended from memory. For every estimate that matters, record 5 things:

• the estimate itself
• where it came from: your own history, an industry base rate or a judgment call
• how many comparable cases sit behind it
• a plausible range, such as 35% to 55% rather than a flat 45%
• what new evidence would move it

“Six of ten similar proposals closed” is a useful observation, and it is still a sample of 10, which guarantees nothing about the next proposal. Uncertainty also lives outside the probabilities. Payoffs, costs, timing and the completeness of the scenario list can all be wrong, so the confidence you place in a calculation should follow the weakest of these inputs.

One shortcut deserves a warning. Multiplying the separate chances that the marketing works, the product works and the timing is right gives the joint probability only when those three are independent, and generally they aren’t, because a product that works tends to make the marketing work too. When the pieces move together, estimate the combined chance directly.

And recalculate when new evidence changes your probabilities or payoffs. A project that started at 60% can be an 80% project after 3 good months or a 40% one after a bad quarter, and the decision to continue should rest on the current number.

Things Money Doesn’t Measure

Expected value isn’t restricted to money. It works on hours, on units, on any outcome you can count. What it can’t do is weigh outcomes by how much they matter to you. A failed launch costs more than its dollar figure if it also costs a year of confidence, and a modest success can be worth more than its revenue if it opens a market. Expected utility is the version of the calculation that handles this. Each outcome passes through a utility function, a score for how much that outcome is worth to you, before it is weighted. It is more complete and more subjective, and it is the formal reason a cautious person can rationally turn down a bet with positive expected value.

Doors That Only Open One Way

Reversible decisions are the natural home of expected value. Try the option, learn, adjust, and the average works in your favor over time. An irreversible decision with a large downside deserves more caution than the number suggests, because a wrong answer can’t be corrected on the next attempt. Jeff Bezos called these one-way doors in Amazon’s 2015 letter to shareholders, and the advice travels well: walk through a two-way door on a good expected value, and stand at a one-way door until the worst case is one you can live with.

A Worksheet for Your Next Decision

A spreadsheet does this arithmetic better than a calculator, because the total updates the moment you change an assumption, which is the whole point of testing one. Put the probabilities in B2:B7 and the net outcomes in C2:C7, then:

=SUMPRODUCT(B2:B7,C2:C7)

SUMPRODUCT multiplies each probability by its outcome and adds the results, which is exactly the expected value formula. Enter probabilities as percentages, 10% rather than 10, and keep a separate cell that totals B2:B7 so you can see at a glance that the model adds up to 100%. The formula works the same way in Excel and in Google Sheets.

If you’d rather start from a finished sheet, the worksheet below has the launch example on its first tab, a break-even tab that works out the threshold and the sensitivity table from your own cost and value figures, a tab for comparing options and an estimate log. Change the yellow cells and everything else recalculates.

Then, before committing to anything, run through this list:

• the expected net result, with every cost counted once
• the chance of losing money, and the worst listed outcome
• whether you could absorb that worst outcome without it changing anything else you care about
• the break-even probability, and what evidence says you clear it
• the best alternative use of the same money and time, because that is the real baseline
• the one assumption that flips the answer if it turns out wrong

If a line stays blank, you have found the thing to investigate before deciding. That is what the calculation is for.

FAQs on Expected Value

Final Remarks

If you’ve worked through the examples, you now have a calculation that takes a few minutes and a set of questions that take longer. The one idea to keep is that expected value makes your assumptions visible. It doesn’t turn uncertainty into certainty.

Before committing to a decision, I want to know the expected net result, the loss I might have to absorb, and which assumptions could change the answer. A positive number is a reason to investigate an option, not permission to ignore its downside.

The worksheet above is yours to keep for the next decision. For practice, the expected value study notes on this site work through more examples, from a project bid to an insurance premium, and end with a 10-question practice set with full solutions. The PDF is free to download from that page.

That’s everything I have on this. I hope it makes your next decision an easier one.