Expected Value, Decision Trees, and How to Decide Under Uncertainty

Last week you bought a lottery ticket, and somewhere a casino owner smiled. Not because you lost-you probably did-but because the math was on their side the moment you handed over the cash. That same math is available to you, and it works just as well for choosing a job, pricing a product, or buying a piece of equipment.

Here’s the uncomfortable truth that runs underneath all of it: a good decision and a good outcome are not the same thing. You decide forward, with patchy information and real uncertainty. Outcomes get judged backward, after luck has had its say. So how do you make the best forward-looking choice you can? With a handful of working tools you can actually pick up and use on Monday morning.

Why this matters

Most of us decide by feel. We weigh a choice in our heads, get a gut reading, and go. That works fine for lunch. It works terribly for anything where the stakes are real and the odds are murky.

The tools here do one quietly powerful thing: they force your hidden guesses out into the open where you can check them. Instead of “this feels like a good idea,” you get “I think there’s a 65% chance, and here’s the number that follows.” You can argue with a number. You can revisit it later and learn. You can’t do either with a feeling.

None of this requires hard math. It requires a little arithmetic and the willingness to write your guesses down instead of keeping them in your head.

First, three flavors of “not knowing”

Every tool in this article helps you decide when you don’t know what will happen. But “not knowing” comes in different strengths, and it matters which one you’re facing.

• Certainty. You know exactly what happens. Press B4 on the vending machine, get the chips.
• Risk. You don’t know the outcome, but you do know the odds. A roulette wheel: you can’t predict the spin, but you know each number’s chance.
• Uncertainty. You don’t know the outcome and you don’t know the odds. Launching a brand-new product into a market nobody has measured-you’re guessing the probabilities themselves. Economist Frank Knight gave this its formal name, which is why it’s sometimes called Knightian uncertainty.

Most of real life is uncertainty wearing a risk costume. The tools below want clean probabilities, but you’ll usually be feeding them estimates. That’s fine, and it’s the whole point. Writing down “I think there’s about a 60% chance” is far better than pretending you know nothing-and far more honest than pretending you know everything.

The takeaway: risk means the odds are known; uncertainty means they’re not. These tools make you name your odds even when you have to guess them. A written estimate you can check later beats a vague feeling every time.

Expected value: putting one number on a gamble

Expected value is the most useful single idea here, and it’s genuinely simple.

Expected value (EV) is the average payoff you’d get if you could repeat a choice many, many times. You compute it by multiplying each possible outcome by its probability, then adding those up.

In plain words: EV = (outcome 1 × its chance) + (outcome 2 × its chance) + …

The lottery ticket, revisited

A ticket costs $2. It gives you a 1-in-10-million chance of winning $5,000,000. What’s the average payoff?

$5,000,000 × (1 / 10,000,000) = $0.50.

So the average ticket is worth about fifty cents, and you’re paying two dollars for it. Across millions of tickets you’d lose three-quarters of your money. The lottery is a beautifully reliable way to turn two dollars into fifty cents.

Flip that around and you understand why casinos and insurance companies are so profitable: they arrange to sit on the positive side of the EV. They don’t win every bet-the casino pays jackpots, the insurer pays claims-but on average, across thousands of bets, the math runs their way. They don’t need luck. They need volume and a positive EV.

A bet whose EV is in your favor is called +EV (or “EV-positive”). One of the most useful mental shifts you can make is to ask of any risky choice: “Is this +EV?” Is it worth taking on average, even though it might lose this particular time?

Common misconceptions

”It lost, so it was a bad bet”

This is the costliest habit in all of decision-making, and it has a name: resulting-judging a decision by its outcome instead of by what you knew when you made it.

Imagine a coin flip that pays $30 on heads and loses $10 on tails. The EV is +$10, so it’s a great bet. If it lands tails and you lose, the bet was still smart. The outcome was just unlucky.

The flip side is equally dangerous: a reckless bet that happens to pay off was still a bad decision. Don’t let a lucky outcome convince you that a sloppy process was wise. Separate the quality of the decision from the quality of the result. They’re different things.

”Expected value tells me everything”

It doesn’t, because pure EV treats every dollar the same-and dollars don’t feel the same in every situation. That’s important enough to get its own section.

Why EV alone can fool you: utility

A million dollars means everything to someone who’s broke and almost nothing to a billionaire. Same dollars, wildly different personal value. The satisfaction a dollar brings depends on how much you already have.

Utility is the personal value an outcome gives you, not its raw dollar amount. Expected utility is just expected value computed in utility instead of dollars: you weight how much each outcome is worth to you by its probability.

Consider a clean choice:

• Option A: a guaranteed $1,000,000.
• Option B: a 50/50 coin flip for $2,000,000 or nothing.

The expected value of both is identical: $1,000,000. Yet almost everyone takes the sure million. Are they being irrational? No-they’re being sensible about utility.

The jump from $0 to $1M changes your life. The jump from $1M to $2M is nice, but it adds far less than the first million did. This is diminishing marginal utility: each extra unit of money matters less than the one before. Because the second million is “worth less” to you, gambling your sure first million to chase it is a bad trade in utility terms, even though it’s an even trade in dollar terms. Preferring the sure thing here is called risk aversion, and for most people it’s perfectly rational.

This idea is old. In 1738, mathematician Daniel Bernoulli introduced utility to crack the St. Petersburg paradox-a coin-flip game whose expected value is mathematically infinite, yet which no sane person would pay more than a few coins to play. The resolution: people don’t value money in a straight line; they value its usefulness, which grows ever more slowly. Measure in utility instead of dollars and the paradox dissolves.

The takeaway: use raw expected value for repeated, low-stakes bets where the amounts are small next to your wealth. For big, one-shot, life-changing stakes, switch to expected utility-ask how much each outcome is worth to you. The million-or-coin-flip choice shows why a “lower-EV” sure thing can be the smarter call.

Decision trees: mapping a choice that unfolds in stages

Expected value handles a single gamble. But real decisions branch: you choose, then chance happens, then maybe you choose again. A decision tree lays the whole thing out so you can compute the EV of each path and pick the best opening move.

Think of it as a choose-your-own-adventure book where, instead of flipping ahead to read what happens, you’ve written the odds on each fork and the prize at the end of each path. Then you trace backward to find which first move gives the best average ending.

A decision tree has three pieces:

• Squares = decision nodes-you choose here.
• Circles = chance nodes-the world rolls the dice; you write the probabilities.
• Payoffs = the value at the tip of each branch.

You solve a tree by folding back (also called rolling back): start at the tips, work leftward, and at each chance node compute its expected value. At each decision node, pick the branch with the highest value. Whatever survives back at the start is your best move.

Example: should you take the startup job?

Say your current job gives you a steady satisfaction we’ll score as 50. A startup offers you a role, and you estimate:

• 60% chance it goes well, payoff 100 (great pay, growth, equity).
• 40% chance it folds within a year, payoff 10 (back to job-hunting).

EV of the startup branch = (100 × 0.60) + (10 × 0.40) = 60 + 4 = 64.

EV of staying = 50.

On these numbers, the startup is the higher-EV move, 64 versus 50. But notice two things. The answer depends entirely on the numbers you put in. And you might still prefer staying if a payoff of 10 would be ruinous for you-that’s utility and risk aversion talking.

The tree doesn’t decide for you. It makes your assumptions visible so you can argue about the right things. Its real power shows up when decisions have multiple stages: choose a strategy, see how the market reacts, then choose again. Drawing it forces you to plan your later moves before you’re emotionally committed, and to spot paths you’d otherwise forget.

The tree’s main gift isn’t the final number-it’s that it drags your hidden assumptions into the open where you and your team can challenge them. A tree built on honest “about 60%” guesses beats a confident decision built on a gut feeling you never examined.

Mental models: the reusable thinking tools

EV and decision trees are specific tools. But good decision-makers also carry a kit of more general thinking tools they reach for across every domain. These are mental models: reusable concepts-borrowed from any field-that help you make sense of a situation. Supply and demand, compound interest, base rates, second-order effects.

Investor Charlie Munger put it best: to a person with only a hammer, every problem looks like a nail. If your one tool is “trust my gut,” you’ll hammer every decision with it, even the ones that need a screwdriver. A full toolbox lets you pick the right tool for the situation in front of you. Three models are useful enough to install right now.

Base rates: start from the outside view

A base rate is the background frequency of something before you look at the specific case in front of you.

Before deciding your startup will surely succeed, note that roughly 90% of startups fail. That 90% is the base rate-the “outside view.” Your detailed, exciting plan is the “inside view.” Both matter, but beginners almost always over-trust the inside view and ignore the base rate entirely.

The fix is an explicit order of operations: start from the base rate, then adjust for what’s special about your case. Never skip straight to the special case. This is exactly how the best forecasters work. Philip Tetlock’s “superforecasters”-ordinary, trained people who out-predicted intelligence analysts in the Good Judgment Project-begin every forecast from the base rate, then update gradually as evidence comes in.

Second-order thinking: “and then what?”

Second-order thinking means tracing the consequences of the consequences, not just the first, obvious effect.

A weak chess player grabs the free pawn-that’s first-order thinking (“this feels good now”). A strong player asks what the free pawn sets in motion three moves later, and notices it leads to checkmate.

Here’s an everyday version. A shop cuts prices to win customers. First-order: more sales this week. Second-order: the rival next door cuts prices too, both shops now earn thin margins, and you’ve started a price war neither can easily exit. The first-order move felt smart; the second-order consequence was a trap. Investors like Howard Marks and Ray Dalio built careers on this insight-the obvious move everyone can see is rarely where the advantage hides.

Reversibility: one-way vs. two-way doors

Not all decisions deserve the same care. A simple model from Jeff Bezos: ask whether a decision is a one-way door (hard or impossible to undo) or a two-way door (easily reversed).

• One-way doors-selling your house, quitting to start a company, a big irreversible spend. Go slow, gather information, build a tree, run a premortem.
• Two-way doors-trying a new supplier for one order, changing a button color, testing a price for a week. Decide fast and learn from the result.

The common mistake is treating every decision as a one-way door. Beginners deliberate for weeks over choices they could reverse in an afternoon, then rush the genuinely irreversible ones. Match the speed of the decision to the cost of being wrong: slow for one-way doors, fast for two-way doors. Agonizing over reversible choices wastes your scarcest resource-your decision-making attention.

How to use this: a worked decision

Watch how the tools stack. Suppose you own a small print shop and you’re deciding whether to buy a second printing machine for $40,000.

1. Frame the uncertainty. You don’t know future demand-this is uncertainty, so your probabilities will be estimates. Name them anyway.
2. Start from the base rate. How often does buying extra capacity pay off for shops your size? From talking to peers, say it’s roughly 50/50. That’s your outside view, before optimism kicks in.
3. Adjust with the inside view. You have a big new client almost confirmed, so you nudge the “demand is high” chance up to 65%.
4. Build the tree and compute EV. If demand is high (65%), the machine earns +$90,000 over its life. If low (35%), you’ve sunk $40,000 for little return, a net of −$30,000. EV = (90,000 × 0.65) + (−30,000 × 0.35) = 58,500 − 10,500 = +$48,000.
5. Check utility, not just dollars. A +$48,000 average looks great-but if losing $40,000 would bankrupt you, that downside hurts far more than its dollar size suggests. Risk aversion is rational when a loss could wipe you out.
6. Apply reversibility. Buying outright is close to a one-way door; leasing turns it into a two-way door. Leasing keeps most of the upside while making the decision reversible and capping the downside.

Notice that no single tool decided this. EV pointed at “yes.” Utility and reversibility refined how to say yes-lease, don’t buy. That’s the realistic picture: these tools don’t replace judgment. They organize it.

Here’s a quick reference for which tool answers which question:

Tool	What it answers	Best used when…
Expected value	What’s the average payoff?	Repeated or small-stakes bets; comparing options on one number.
Expected utility	What’s it worth to me?	Big, one-shot stakes where a loss could hurt badly.
Decision tree	What’s my best move across stages?	Choices that unfold over time, with chance steps in between.
Base rate	What usually happens?	Always-start here before the specifics.
Second-order thinking	And then what?	Any move others will react to, or with delayed effects.
Reversibility test	How carefully should I decide?	To set the speed of any decision.

The limits of the tools (and why that’s okay)

It’s tempting to think that with enough tools you can calculate your way to certainty. You can’t, for three honest reasons.

Your inputs are guesses. A tree built on numbers you invented is only as good as those numbers-garbage in, garbage out. The tree doesn’t manufacture truth; it makes your guesses explicit and checkable. That’s still a huge gain, but it isn’t magic.

More information isn’t always better. Past a point, extra data adds noise and false confidence rather than accuracy-you feel more certain without being more right. Economist Herbert Simon called the realistic alternative bounded rationality: with limited time and brainpower, we satisfice, picking the first option that’s good enough rather than chasing the mathematically perfect one. A good habit: decide what facts would actually change your mind before you go hunting. If a piece of data wouldn’t change your choice, you don’t need it.

Outcomes still belong to luck. Even a perfectly built +EV decision can lose this one time. The tools improve your process, and the process is the only thing you actually control. The outcome is process plus luck-and luck isn’t yours to command.

Conclusion

If you remember one thing, make it this: the process is the product. You can’t control whether any single decision turns out well. You can only control the quality of how you decided. Expected value, utility, decision trees, and mental models are how you raise that quality-and over many decisions, a sound process wins even though good decisions sometimes lose.

Which raises the obvious next question. If these clean tools are this good, why do smart, well-meaning people ignore them so reliably? Because the human mind comes pre-loaded with predictable biases that quietly pull every real decision away from the math. Knowing the tools is half the battle. Knowing how your own brain sabotages them is the other half-and that’s where we go next.