The Difference Between Investing and Gambling Is One Number

A photo I took of the pull tab boxes at the bar next to where I live.

A few months ago I was at a bar with some friends. One of them asked the bartender for pull tabs. They paid them several dollars and the bartender gave them pull tabs from the large box behind the bar, pulling tabs with the enthusiasm of someone who had just discovered a retirement strategy. I watched for a minute and then, I thought to myself:

“This is honestly a terrible bet.”

You might be wondering why? Simply put, if you bought the entire box you’d lose money.

This is the part where most people think, “What do you mean the entire box?” So let me explain, because this small moment at a bar is actually the cleanest possible illustration of one of the most important ideas in finance.

What Is a Pull Tab?

Pull tabs are little paper tickets, sold in sealed boxes, where each ticket has a hidden combination of symbols. Some combinations win money. Most don’t. The key detail is that every box has a fixed, predetermined set of outcomes. The total amount paid into the box always exceeds the total amount paid out. If a box costs $500 to buy out completely, maybe $350 in prizes are distributed across it. The rest goes to the bar and the game operator. This is not a wild guess. In Washington State, where pull tabs are regulated, operators must pay out at least 60% of receipts in prizes, and the actual statewide commercial average runs around 73% [1]. Either way, the house keeps the difference, every single time.

This means that no matter how lucky you feel, no matter how many times you have won before, the aggregate result of playing pull tabs is that you lose money. Not eventually, not under bad luck. Mathematically, guaranteed, over enough plays. This is what it means for a system to have negative expected value.

Expected Value: The One Number That Matters

Expected value is the average outcome of a bet or decision if you were to repeat it a very large number of times. Here is the simplest way I know to think about it.

Imagine a jar with 50 green marbles and 50 red marbles. Every time you draw a green marble, you win a dollar. Every time you draw a red marble, you lose a dollar. Every time you draw a marble you put it back into the jar and shake the jar. The expected value of a single draw is exactly zero. Half the time you win a dollar, half the time you lose a dollar, and over many draws you end up right where you started.

Now change the jar so it has 60 green marbles and 40 red marbles. Suddenly the expected value of each draw is positive. On average, you gain twenty cents every time you reach in. Play this game a thousand times and you will be meaningfully ahead.

Flip the ratio and put 40 green marbles and 60 red marbles in the jar. Now the expected value is negative. Every draw costs you twenty cents on average. This is what every casino game, every pull tab box, and most lottery tickets look like from the inside(or worse).

The central argument of this article is this: the meaningful distinction between investing and gambling is not about risk, volatility, or respectability. It is about whether the expected value of the system you are participating in is positive or negative.

An interactive marble-jar widget for exploring how expected value changes as the mix of winning and losing outcomes changes.

The Stock Market as a Positive Expected Value System

Think back to the marble jar. Every year you hold the S&P 500 is essentially one draw from that jar. You do not know the exact composition — no one does — but you can approximate it the same way you would estimate any expected value: by looking at the average outcome across a large number of draws. Over roughly a century of data, the S&P 500 has returned approximately 10% per year on average in nominal terms, measured as a geometric (compound annual) total return with dividends reinvested [2]. That is your best estimate of the expected value of a single year in the jar.

And the draws have been consistent enough to be striking. Over nearly every rolling 20-year window in the data I examined, investors in the S&P 500 made money on a total-return basis [2], [3]. The handful of exceptions occur around windows beginning near the onset of the Great Depression, where returns were slightly negative depending on the exact start date and calculation method. [3]. The claim even holds after adjusting for inflation: the weakest 20-year real return, ending in the early 1980s after a decade of stagflation, was still slightly positive at under 1% per year. The one thing this consistency depends on is reinvesting dividends. On price alone, some long stretches come much closer to breaking even.

This is not an accident. When you buy a broad index fund, you are buying fractional ownership of hundreds of productive companies. Those companies employ people, create products, and generate revenue. In aggregate, over time, they grow. The expected value of each draw is positive because the underlying system — the productive capacity of the broad economy — is structurally tilted toward growth.

What makes positive expected value systems especially powerful is that compounding means each draw is taken from a balance that already includes all the previous wins. A 10% annual return does not just add 10% to your balance each year — it adds 10% to a balance that is already larger than where you started. Over long time horizons this turns modest annual returns into something that looks almost unreasonable. A $10,000 investment compounding at 10% annually is worth roughly $67,000 after 20 years and around $175,000 after 30. You did not work for that difference. The math did. And the more draws you accumulate, the more reliably your observed results converge toward the underlying expected value — which is precisely why the 20-year windows are so consistent, and why time is the single most powerful variable in the equation.

This makes it all the more counterintuitive that most people who try to time the market often end up worse off than if they had simply held it [16].

I made a widget where you can explore approximate 20-year S&P 500 windows yourself, including price-only returns, a simplified dividend assumption, and inflation-adjusted returns.

When Positive Expected Value Goes Wrong: Day Trading

Here is where it gets uncomfortable for a lot of people.

Many people look at the stock market and think the right approach is to trade it actively, jumping in and out of positions, trying to time every move. The data on this approach is brutal. A landmark study of individual investors found that the households that traded the most earned the lowest returns, badly trailing the market [4]. A separate study covering the entire population of day traders on the Taiwan Stock Exchange found that fewer than 1% were able to reliably earn positive returns net of fees [5]. And a study following day traders in the Brazilian equity futures market found that 97% of those who persisted for more than 300 days ended up with net losses [6].

How can the stock market have positive expected value but day trading within it tend toward negative expected value? The answer is friction. Transaction costs, bid-ask spreads, taxes on short-term gains, and the difficulty of consistently out-predicting markets that are adversarial all add up.

This does not mean that systematic, rules-based trading is impossible to do profitably. There are well-documented market anomalies, including momentum effects [7] and factor-based strategies built on characteristics like size and value [8], that have historically shown edge. But these require rigorous quantitative work, not intuition and a stock screener. The people who find genuine edge in markets tend to think about it the way a scientist would: form a hypothesis, test it against historical data, stress-test the assumptions, apply it with discipline and monitor live system telemetry. The key difference is that they are solving for expected value, not excitement.

Prop Firms: The Casino Hiding in Plain Sight

In recent years a type of business has exploded in popularity targeting aspiring traders: proprietary trading firms, or prop firms(also sometimes called challenge firms). The pitch is seductive. Pay a fee to take a challenge, prove you can trade profitably under their rules, and they will fund you with their capital and you keep a cut of the profits.

The reality is more interesting. Prop firms tend to impose rules specifically designed to be difficult to meet: no holding positions overnight, no running systematic or algorithmic strategies, mandatory daily loss limits that are extremely tight. These constraints are not designed to identify great traders. They are designed to ensure that most participants fail, and to make it structurally difficult to operate in a style that has historically produced edge.

The business model, stripped down, looks like this. Charge everyone an assessment fee. Most people fail the challenge. Of those who pass, give them a funded account with rules that make long-term profitability unlikely. Collect a share of any profits from the rare few who succeed. The prop firm is not betting on your success. It is betting on your failure, because statistically that bet wins most of the time.

In expected-value terms, this starts to resemble a casino. The house edge comes not from the cards or the wheel but from the fee structure and the rules of engagement. Like any casino, a prop firm does have some winners. But on aggregate, across all participants, it is a system with negative expected value for the traders and positive expected value for the firm. That is precisely how the firm stays in business.

Casinos: At Least They Are Upfront About It

At least casinos are transparent about their edge.

Every casino game is designed with a house advantage built in. American roulette carries a house edge of about 5.26%, meaning for every dollar bet the casino expects to keep around five cents on average [9]. Slot machines typically run house edges of roughly 5% to 10%, depending heavily on the denomination and jurisdiction, with low-denomination penny machines running near the high end and high-denomination machines near the low end [10]. Even blackjack, which with perfect basic strategy can reduce the house edge to around 0.5% under good rules, is still designed to favor the house in aggregate, and that edge climbs sharply under worse rules like a 6:5 blackjack payout [11].

The interesting exception in blackjack is card counting. A skilled player who tracks the ratio of high cards to low cards remaining in the shoe can determine when the deck composition favors the player and bet larger in those moments. Card counting does not guarantee winning any individual hand, but over enough hands it flips the math. The expected value of the game shifts from negative to positive for the counter [12].

And here is where things get socially interesting. In finance, someone who develops a systematic, quantitative edge over the market is called a quant, a systematic trader, or a fund manager, and they are celebrated for their analytical rigor. In a casino, someone who does the mathematical equivalent in blackjack is labeled a card counter, treated as a quasi-criminal, and banned from the premises. The underlying act is identical: finding and exploiting a statistical edge in a system. The social reception is completely different, and it says more about institutional interests than about anything intrinsically distinct between the two activities.

Why Finding Positive Expected Value Is Hard

If positive expected value is what separates investing from gambling, a natural question follows: why doesn’t everyone just go find positive expected value opportunities? This is a good question. Markets are sometimes described as “predictably random,” and I think that framing is right if you put the emphasis on the word predictably. The randomness is real and humbling, but it is not perfectly uniform. Factor-based strategies, momentum effects [7], [8] and mean-reversion tendencies, where past losers have historically tended to outperform past winners over longer horizons [13], have all shown up persistently enough to suggest that the market is not purely random noise.

It should also be noted that these effects operate under different conditions and time horizons. Momentum and mean reversion in particular pull in opposite directions, and determining which dominates in a given context is a difficult empirical problem.

The question is whether your method of finding and exploiting those patterns is rigorous enough to overcome transaction costs and the inevitable periods of underperformance. That is a genuinely difficult problem, but it is a solvable one.

You Do Not Need to Know the Exact Number

One reasonable objection at this point: expected value sounds useful in theory, but how do you actually calculate it when you are dealing with something as complex as a trading strategy or a business investment?

For simple systems like pull tabs or roulette, the math is exact and someone has already done it for you. For more complex systems, the honest answer is that you usually cannot know the expected value precisely. What you can do is approximate it, and often a good approximation is enough to act on.

This is what backtesting does. You apply a set of rules to historical data and observe how they would have performed. The result is an estimate of the strategy’s expected value, subject to assumptions and with known limitations. Strategies that look great in backtests do not always perform as well in live markets. But an imperfect estimate that tells you “this system wins about 55% of the time and the average win is meaningfully larger than the average loss” is actionable, even if you cannot pin down the exact number with certainty.

The same principle applies to most real-world financial decisions. You rarely know exactly what a business investment will return, but you can reason about whether the expected value is likely positive or negative based on available evidence. Precision is useful. Direction matters more.

Variance, the Law of Large Numbers, and Why Time Is Your Friend

There is one more concept worth addressing carefully, because it trips people up.

Positive expected value is not a guarantee of success in the short run. This is where the law of large numbers becomes important. The law of large numbers is a theorem in probability that says, as the number of trials increases, the observed average of your outcomes converges toward the expected value. In plain terms: the more times you play a positive expected value game, the more likely your actual results are to reflect the underlying math.

Think back to the marble jar with 60 green marbles and 40 red ones. On any single draw, you might pull red. Over ten draws, you might still be down. But over ten thousand draws(with replacement), it becomes extremely unlikely that you are not ahead. The positive expected value of the system expresses itself more and more reliably as you accumulate trials.

This is why the 20-year SPY windows are so consistent. A single year in the stock market can be brutal: 2008, 2020, and 2022 all saw major drawdowns. But over 20 years, the law of large numbers has had time to work. Your observed return converges toward the underlying expected return of the system.

The same logic applies to any systematic trading strategy with genuine positive expected value. In the short run, variance dominates. A good system will have losing days, losing weeks, and sometimes losing months. None of that necessarily means the system is broken. It means you have not yet accumulated enough trials for the expected value to fully express itself in your observed results.

This is also why position sizing matters so much. Variance in the short run can be large enough to wipe you out before the law of large numbers has a chance to help you. If you size your bets too aggressively, a bad streak early on can end the game before the math ever gets to work in your favor. The Kelly Criterion is a mathematical formula for the theoretically optimal fraction of your capital to risk given your edge and the odds [14]. In practice, the inputs to Kelly are uncertain since you rarely know your true edge precisely, and the output can vary widely depending on your assumptions. A reasonable alternative is to work backward from your maximum tolerable drawdown: estimate the worst expected drawdown for a given percentage of capital deployed, and size your position so that drawdown stays within what you can absorb. If a system has an expected maximum drawdown of 10% and you have $100 in capital but cannot tolerate losing more than $3, you cap your position at $30 deploying only the fraction of capital where the math works within your constraints.

There is one important caveat to all of this. The law of large numbers assumes the underlying distribution stays stable across your trials. In markets, that assumption is not always safe. Edges erode as other participants discover and exploit the same inefficiencies. Macro regimes shift. A strategy with strong historical expected value can see that value decay if the conditions that generated it change. This is actually one reason the SPY example is a stronger argument than most specific trading strategies: the underlying source of return is broad economic growth and corporate earnings, which is a more structurally durable foundation than most specific market anomalies. The more niche your edge, the more vigilant you need to be about whether it is still intact.

The Gambler’s Fallacy: A Dangerous Misreading of the Same Math

The law of large numbers is one of the most useful ideas in probability. It is also one of the most commonly misunderstood, and that misunderstanding has a name: the gambler’s fallacy.

The gambler’s fallacy goes like this. You have been on a losing streak. You have lost five times in a row. Surely, the reasoning goes, you are due for a win. The universe needs to balance things out. The so-called law of averages demands it. So you keep playing.

This is objectively wrong, and understanding why requires separating two things that are easy to conflate. The law of large numbers says that averages converge toward expected value over many trials. It says nothing about what happens on the next individual trial. Each pull of a slot machine, each spin of a roulette wheel, each pull tab in the box is an independent event. The machine does not know you have been losing. It has no memory. The probability of winning on the next play is exactly the same as it was on the first play.

The confusion arises because people apply aggregate logic to individual events. Yes, over thousands of spins a roulette wheel will converge toward its expected distribution of outcomes. But that convergence does not come from future wins compensating for past losses. It comes from the sheer accumulation of independent trials, each one carrying the same unchanged odds.

Here is the part that makes the gambler’s fallacy especially costly in a negative expected value system. Even if a losing streak genuinely is followed by a winning streak, your wins will still be smaller than your losses in aggregate. The law of large numbers is not your friend at the roulette table. It is working against you, steadily converging your results toward the negative expected value of the game. The longer you play, the more certain your losses become, not less. A hot streak is not a recovery. It is a temporary detour on the way to the house’s guaranteed cut.

The practical implication is simple. No streak, hot or cold, changes the math of the system you are operating in. If the jar has more red marbles than green ones, it has more red marbles than green ones on every single draw, regardless of what the last ten draws looked like.

Why People Gamble Anyway

To be fair to everyone at the pull tab machine: expected value is not the only thing that matters to human beings.

Gambling provides entertainment. The near-miss, the possibility of a sudden large win, the social atmosphere: these have real value to people. Intermittent, unpredictable rewards are among the most psychologically potent reinforcement patterns we know of [15]. There is nothing wrong with spending $20 on pull tabs the same way you would spend $20 on a movie ticket, as long as you are clear that you are paying for an experience and not building wealth.

The problem arises when people mistake negative expected value systems for positive ones, when they believe that luck, intuition, or a hot streak has changed the underlying math. It has not. The jar still has more red marbles than green ones. Knowing that does not mean you cannot enjoy the game. It just means you know what you are playing.

The Takeaway

Most significant financial decisions can be usefully framed as questions about expected value. Can I approximate the expected value of the system I am operating in well enough to act on it? If I believe the expected value is in my favor, do I have enough trials ahead of me for the law of large numbers to do its work?

The tools for answering those questions range from simple arithmetic to rigorous quantitative modeling, but the underlying logic is the same whether you are thinking about an index fund, a trading strategy, a prop firm challenge, or a pull tab machine in a dive bar.

Know what game you are playing. Know which way the math tilts. And if you are not sure, at minimum know that you are not sure.

If you want to see what building around positive expected value actually looks like in practice, I wrote a 3-part series on the infrastructure behind a production trading system — starting with ETFs Aren’t Magic. Here’s What’s Inside One.

References

[1] Washington State Legislature, “WAC 230–14: Punch boards and pull-tabs.” [Online]. Available: https://app.leg.wa.gov/wac/default.aspx?cite=230-14&full=true

[2] A. Damodaran, “Historical returns on stocks, bonds and bills: 1928–current,” NYU Stern School of Business. [Online]. Available: https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/histretSP.html

[3] “S&P 500 historical return calculator,” DQYDJ. [Online]. Available: https://dqydj.com/sp-500-historical-return-calculator/

[4] B. M. Barber and T. Odean, “Trading is hazardous to your wealth: The common stock investment performance of individual investors,” The Journal of Finance, vol. 55, no. 2, pp. 773–806, 2000.

[5] B. M. Barber, Y.-T. Lee, Y.-J. Liu, and T. Odean, “The cross-section of speculator skill: Evidence from day trading,” Journal of Financial Markets, vol. 18, pp. 1–24, 2014. [Online]. Available: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=529063

[6] F. Chague, R. De-Losso, and B. Giovannetti, “Day trading for a living?” SSRN Working Paper 3423101, 2020. [Online]. Available: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3423101

[7] N. Jegadeesh and S. Titman, “Returns to buying winners and selling losers: Implications for stock market efficiency,” The Journal of Finance, vol. 48, no. 1, pp. 65–91, 1993.

[8] E. F. Fama and K. R. French, “The cross-section of expected stock returns,” The Journal of Finance, vol. 47, no. 2, pp. 427–465, 1992.

[9] M. Shackleford, “Roulette,” Wizard of Odds. [Online]. Available: https://wizardofodds.com/games/roulette/basics/

[10] Nevada Gaming Control Board, “Gaming Revenue Report: slot win and hold percentages,” 2024. [Online]. Available: https://gaming.nv.gov/about/gaming-revenue/information-and-reports/

[11] M. Shackleford, “Blackjack house edge calculator,” Wizard of Odds. [Online]. Available: https://wizardofodds.com/games/blackjack/calculator/

[12] E. O. Thorp, Beat the Dealer: A Winning Strategy for the Game of Twenty-One. New York, NY, USA: Random House, 1962.

[13] W. F. M. De Bondt and R. Thaler, “Does the stock market overreact?” The Journal of Finance, vol. 40, no. 3, pp. 793–805, 1985.

[14] J. L. Kelly Jr., “A new interpretation of information rate,” The Bell System Technical Journal, vol. 35, no. 4, pp. 917–926, 1956.

[15] C. B. Ferster and B. F. Skinner, Schedules of Reinforcement. New York, NY, USA: Appleton-Century-Crofts, 1957.

[16] I. D. Dichev, “What are stock investors’ actual historical returns? Evidence from dollar-weighted returns,” American Economic Review, vol. 97, no. 1, pp. 386–401, 2007.

Note

None of this is financial advice; it is a framework for thinking about risk, expected value, and incentives.