The Rain Seller: What an Umbrella Can Teach Us About Variance in Poker

Imagine that you sell umbrellas. You have a good product, a competitive price and you know how to serve your customers. You chose a good location, you prepared the premises and you do everything that depends on you. However, it doesn't rain for weeks and sales are barely enough to cover part of the expenses.

A few blocks away, another vendor opened his business just before a storm. He has less experience and his umbrellas are worse, but he sells everything.

If you only looked at that week's collection, you could conclude that he knows how to sell and you don't. You would be confusing the quality of the business with the conditions that affected it.

Something similar happens in poker. You can make good decisions and lose for a period. You can also make mistakes and win. The result contains information, but by itself it is not enough to explain how you played.

The analogy has a second part: perhaps you can't decide when it rains, but you can choose where to set up the business. And that choice changes how much you depend on the next storm.

Earning on average does not mean earning regularly

To understand it, let's get out of poker for a moment. Suppose you have a choice between two games. In both you pay USD 10 per attempt.

In game A, half of the time you win USD 10 net and the other half you lose USD 10. In game B, nine times out of ten you lose USD 10, while when you are correct you win USD 90 net.

The expected value is the average of the possible outcomes, weighted by their probabilities.

For A:

EV = 0.50 × 10 + 0.50 × (−10) = USD 0.

For B:

EV = 0.10 × 90 + 0.90 × (−10) = USD 0.

The two games have the same expected value. However, the experience of playing them is very different. In A, the payouts are frequent and small. In B, the money arrives in a much more concentrated way.

This is where the variance appears: the dispersion of the results around the expected value.

Having the same expected average does not imply having the same distance.

Put a number on the dispersion

The variance is calculated by taking the distance of each result from the average, squaring it, and weighting it by its probability. Squaring prevents positive and negative differences from canceling out.

Since both games have zero expected value, the accounts look like this:

Variance of A = 0.50 × 10² + 0.50 × (−10)² = 100 USD².

Variance of B = 0.10 × 90² + 0.90 × (−10)² = 900 USD².

Game B has nine times the variance of A.

To express this dispersion in dollars, we can use the standard deviation, which is the square root of the variance.

The standard deviation of B is three times greater. It doesn't mean you'll always end up that far from the average, nor does it set a loss limit. It is a measure of how much the results are dispersed.

The difference has concrete consequences: although both games have the same expectation, they may require very different reserves to get through unfavorable periods.

Ten attempts without collecting: exceptional in one, frequent in the other

Suppose that the trials are independent and that the probabilities do not change. The probability of failing ten times in a row is obtained by multiplying the probability of failing ten times.

In game A:

0,50¹⁰ ≈ 0,098%.

In game B:

0,90¹⁰ ≈ 34,87%.

In a block of ten attempts, not cashing out at all is very rare in A. In B it happens in more than a third of blocks of that size.

If we extend the block to twenty attempts, the probability of not collecting in B remains:

0,90²⁰ ≈ 12,16%.

In that case you would have lost USD 200 without obtaining a single payment. It wouldn't have changed the game or made his odds any worse. You would be going through one of the possible routes of the distribution you chose.

These accounts describe a block defined before starting. They do not represent the probability of finding such a streak within thousands of attempts: having many opportunities for it to appear increases that probability.

Nor do they mean that, after twenty failures, the prize is close. If the attempts are independent, the probability of getting the next one correct is still 10%.

The game does not keep track of what it owes you.

Two cities, the same business and different reservation needs

Let's go back to the umbrella seller. It can be installed in two hypothetical cities.

In the first one it rains frequently and the sales are spread over many days. In the second there are prolonged dry periods and storms in which there is a lot of sales. Suppose that, considering sales and costs, both businesses leave the same expected annual profit.

Does it matter where to open?

If the seller has ample reserves and can go through slow months, perhaps either of the two alternatives is viable. But if you need to withdraw money every month to live, income distribution changes your situation.

The rent does not wait for the storm to arrive. Expenses appear even if sales do not.

In poker, the bankroll fulfills part of that function: it allows you to continue playing during unfavorable periods. If you also need to withdraw money, the account must absorb both gambling losses and living expenses.

A business can have a positive expectation and run out of resources before a big result arrives. Therefore, in addition to wondering how much you could earn, you need to understand if you can sustain the journey.

In MTTs, the variance is also in the prize structure

When we talk about variance, we usually think of flips, bad beats and cards that don't match. All of this influences, but does not completely describe the distribution of results of a tournament.

An MTT combines many possibilities: being left out without being paid, recovering part of the investment through an bounty, entering prizes, reaching an final table or getting one of the first positions. Each outcome has a different probability and return.

Therefore, just looking at how many all-ins you won is not enough to explain the variation in your financial results. It also matters how much money is focused on less frequent outcomes and how much of your expected return depends on reaching them.

You can win many pots without reaching the prizes that weigh the most. And you can have an extraordinary period because one of your best performances coincided with the biggest tournament on your tournament schedule.

The payout frequency does not resolve the entire question either. Two tournaments can pay the same percentage of participants and distribute the money in very different ways. If one concentrates a greater proportion of the prizes in the first positions, their economic structure changes even if both have the same percentage of ITM.

Same expected ROI, two ways to get paid

Now let's modify the first example so that both games have positive expectation.

Each entry still costs $10, but A pays $24 with a 50% probability and B pays $120 with a 10% probability. If you don't get it right, you charge zero.

These are simplified models, not actual tournament structures. They serve to isolate the effect of the distribution of payouts.

The expected profit of A is:

0.50 × 14 + 0.50 × (−10) = USD 2.

The one from B:

0.10 × 110 + 0.90 × (−10) = USD 2.

In both cases, USD 2 of expected profit over USD 10 of cost is equivalent to an expected ROI of 20%.

But the dispersion is still different. To calculate it, we subtract the average of USD 2 from each result:

Variance of A = 0.50 × (14 − 2)² + 0.50 × (−10 − 2)² = 144 USD².

Variance of B = 0.10 × (110 − 2)² + 0.90 × (−10 − 2)² = 1,296 USD².

The standard deviations are USD 12 and USD 36, respectively.

Now let's imagine 1,000 entries. In both models you invest USD 10,000 and your expected profit is USD 2,000.

If the entries are independent and maintain the same probabilities, the standard deviation of the accumulated result is calculated by multiplying the deviation per entry by the square root of the volume:

Model A: 12 × √1,000 ≈ USD 379.47.

Model B: 36 × √1,000 ≈ USD 1,138.42.

Same volume, same cost and same expected ROI. However, model B has three times the dispersion.

These values ​​are not profit or loss limits. They also do not allow loss percentages to be transferred to real tournaments without knowing their distributions. What they show is something more basic: the expected ROI, by itself, does not describe the economic experience of playing a tournament schedule.

Volume helps, but it does not force the result to compensate you

In these independent models, the total expected profit grows in proportion to the number of entries. The total standard deviation grows in proportion to its square root.

If you multiply the volume by four, you multiply the expected profit by four and the standard deviation by two. Uncertainty becomes lower relative to expectation, although dispersion measured in dollars increases.

That explains why more volume can help estimate an average. It does not mean that there is a number of tournaments after which you are guaranteed to win, nor that a bad streak generates pending favorable results.

Additionally, adding volume only preserves this logic if you maintain the model conditions. In practice, adding tables or extending sessions can modify the quality of your decisions. If your advantage goes down, your expected value also changes.

Playing more is not simply repeating the same game more times when fatigue makes you play it differently.

Your tournament schedule also chooses a climate

Choosing a tournament means choosing more than just an buy-in. You also choose a field size, a structure, a prize distribution and an environment in which your advantage may be greater or lesser.

Smaller fields and less concentrated prizes can reduce dependence on exceptional results. But the size of the field, isolated, does not allow us to calculate the variance or decide which tournament is best. The level of rivals, total cost, speed, reentries and the full distribution of returns, including bounties where applicable, also matter.

Reducing variance does not guarantee increasing profitability either. You can find a more stable tournament in which you have less advantage and another more dispersed one with a higher expectation. To compare them, it is necessary to look at both things, in addition to the time each one requires.

The decision also has a personal dimension: how much capital you can allocate, what volume you can sustain and how much you need to withdraw.

The same tournament schedule can be bearable for someone with ample reserves and external income, and very difficult to sustain for someone who depends on poker withdrawals to pay their expenses.

Your tournament schedule also chooses a weather. That choice changes the economic distribution to which you are exposed.

Variance does not certify that you are playing well

Understanding all this helps to avoid hasty conclusions. A bad period does not in itself prove that you are a loser. Nor does a big win on its own prove that you have a sustainable advantage.

But there is a catch: using variance as an automatic explanation for any negative results.

The above accounts work because we define the probabilities and expected value. In real poker, precisely, one of the things we try to estimate is our advantage.

Believing that you have an ROI of 20% does not make it a known fact. There may be technical errors, poor tournament selection, or more difficult opponents than you expected. There can also be good decisions accompanied by bad results.

That is why it is advisable to look at the process in addition to the balance: review hands, compare decisions with technical criteria and evaluate whether the conditions under which you estimated your advantage are still valid.

The seller needs to recognize when they are going through a dry season. You also need to detect if your umbrellas have holes.

Can you wait until the next storm?

The umbrella seller needs to know how to sell. You also need to understand the climate of the city you work in and have resources to get you through dry days.

The same thing happens in tournaments. Skill matters, but the economic path also depends on how the prizes are distributed and how much your expectation depends on the less frequent results.

You cannot program an final table. Yes, you can review how much you invest, what tournaments you choose and what margin you have to continue playing if that prize takes much longer than you imagined.

You can't make it rain. But you can choose in which weather to sell umbrellas.

The question is not just how much you could earn when the storm hits.

It is whether you can sustain your tournament schedule until the next storm.

All numerical examples are hypothetical. The block and cumulative dispersion calculations assume independent trials, constant probabilities and a fixed cost. Payouts are gross; net profit subtracts the stated cost. They do not estimate the bankroll requirements or risk of ruin of a real player.