Learning goals and lesson guide
**Estimated reading time:** 8 min read.
**Level:** Intermediate. **Study time:** Allow 35–45 minutes with a calculator and paper ledger. **Prerequisites:** Expected value, probability and net returns. All model inputs below are invented and assume no real-money activity.
A model has a positive expected result under its assumptions, yet a short paper sequence loses money. That observation does not automatically prove the model is wrong, but neither does the phrase “just variance” prove it is right. You need to distinguish random variation from poor assumptions and examine both.
This lesson introduces variance as the spread of possible outcomes around an expectation. You will calculate a simple example, distinguish a specified losing sequence from a run occurring anywhere, and measure a paper drawdown. You will also learn why a longer record does not remove model risk or justify increasing exposure after losses.
## 1. Expectation does not specify a path
Consider independent fictional trials with a 55% win probability, decimal odds of 2.00 and a fixed one-unit stake. Each net result is +1 or −1. Expected net per trial is 0.55 − 0.45 = +0.10 units.
Over 100 such trials, expected total net is ten units. That is not a promise that the 100th trial finishes at ten or that the path rises smoothly. A sequence can begin with several losses, move above expectation or finish below zero.
The assumptions are strong: the same probability, same price and independence across trials. Real sports records may violate all three. Use this model to understand arithmetic, then label the ways it differs from a real dataset.
## 2. Calculate variance for one trial
Variance is the probability-weighted squared distance from the mean. Here the mean is 0.10. A win is 0.90 above it and a loss is −1.10 below it. Squaring and weighting gives 0.55 × 0.90² + 0.45 × 1.10² = 0.99.
The standard deviation is the square root of variance, approximately 0.995 units. Its size compared with the 0.10-unit expectation shows how noisy one result is relative to the average implied by the model.
Squaring prevents positive and negative deviations from cancelling. Standard deviation returns the measure to the original units. You do not need to memorise a formula without meaning: list possible net results, calculate their distances from the mean and weight their squared distances.
## 3. Combine independent trials carefully
For 100 independent trials of the same kind, variances add to 99 and total standard deviation is about 9.95 units. Expected total is ten. The standard deviation is a measure of spread, not a guaranteed maximum loss or a complete probability interval by itself.
For the average result per trial, standard deviation is about 0.0995 units. With more independent observations, the average can become more stable even though the absolute total still varies. This distinction explains why a larger study can improve estimation without making the path harmless.
If outcomes are correlated, simply adding individual variances omits covariance terms. Several selections linked to the same match or lineup can move together. A record of 100 rows does not necessarily contain 100 independent pieces of information.
## 4. A specified streak is not a streak anywhere
Under the fixed independent model, the probability that the next five trials all lose is 0.45⁵, approximately 1.85%. That answers a specific question about a particular block of five trials.
The chance of encountering at least one five-loss run somewhere in 100 trials is different and larger. There are many possible starting positions, and overlapping runs are dependent. Do not multiply 1.85% by the number of positions and present the result as an exact probability; overlaps complicate the calculation.
Simulation or an appropriate recurrence can estimate the anywhere-in-the-sequence probability. The advanced course revisits simulation. For now, state precisely which streak event you have calculated. A correct number attached to the wrong question is still a misleading result.
## 5. Order matters for drawdown
Two sequences can have the same final total but different paths. Four wins and four losses at even money finish at zero with fixed one-unit stakes. WWWWLLLL reaches +4 before falling to zero; WLWLWLWL repeatedly returns to zero after smaller peaks.
Define a running cumulative net result, the highest cumulative value reached so far and drawdown as that high-water mark minus the current cumulative value. Include the initial zero as a possible peak. Maximum drawdown is the largest such difference within the studied period.
| Step | Net result | Cumulative | Running peak | Drawdown | | --- | --- | --- | --- | --- | | Start | — | 0 | 0 | 0 | | 1 | +1 | 1 | 1 | 0 | | 2 | −1 | 0 | 1 | 1 | | 3 | −1 | −1 | 1 | 2 | | 4 | +1 | 0 | 1 | 1 |
The final result is zero, yet maximum drawdown is two units. A final profit figure alone hides this path information. A historical maximum drawdown is also not a ceiling on future drawdown.
## 6. Stake size changes monetary variation
If every fixed hypothetical stake doubles, each net result doubles. Expected net and standard deviation double, while variance multiplies by four. The probability of the sporting outcomes does not improve.
This is why increasing a stake after losses should not be described as reducing uncertainty. It changes the amount exposed to the next outcome. A recovery target is an accounting desire, not evidence about the next event.
Keep this lesson's paper stakes fixed so you can observe the sequence without mixing in changing-size effects. Later research can model alternative policies as separate simulations, but no policy should be presented as making a negative or uncertain process safe.
## 7. Distinguish outcome noise from model error
Outcome noise means variation even if the assumed probabilities were correct. Model error means those probabilities or other assumptions may be wrong. Both can operate simultaneously. A poor run might be ordinary variation, a changed environment, a data error or a combination.
Review whether the inputs were available at the decision time, whether market terms were consistent and whether the population changed. A roster change can make older observations less relevant. A coding error can produce a misleading expected value. Neither is repaired by waiting for more results from the same flawed process.
Avoid two extremes: abandoning a method after every short loss sequence, or defending it indefinitely by calling every failure variance. Define review checkpoints and diagnostic criteria before looking at the next batch. This reduces the temptation to explain every outcome retrospectively.
## 8. A paper comparison exercise
Create two eight-trial sequences with four wins and four losses at 2.00, one alternating and one with all wins followed by all losses. Calculate cumulative results and maximum drawdown for each. Explain why equal final totals do not imply identical experience.
Then repeat the accounting using two units per trial. The final total remains zero, but peak-to-trough movements double. This demonstrates exposure scaling without changing the sporting sequence.
Finally, write the assumptions needed to use the 0.45⁵ calculation. If win probabilities differ from trial to trial but independence is retained, a particular five-loss probability becomes the product of the five separate loss probabilities. If independence is not justified, that simple product is not sufficient.
## 9. Review questions and explained answers
1. What is expected net per trial for p = 0.55 at odds of 2.00? 2. Does that expectation promise a positive next result? 3. What does 0.45⁵ calculate under the stated model? 4. Why is it not the exact chance of a five-loss run anywhere in 100 trials? 5. A cumulative record peaks at +6 and later falls to +1. What is the drawdown? 6. If fixed stakes triple, how do standard deviation and variance scale? 7. Can every poor run be dismissed as ordinary variation?
### Answer key
1. +0.10 units per unit staked. 2. No. The next result is still +1 or −1 in this model. 3. The chance that a specified set of the next five independent trials all lose when each loss probability is 0.45. 4. The longer sequence contains many overlapping opportunities for a run; those events are not independent. 5. Five units, despite the cumulative result still being positive. 6. Standard deviation triples and variance multiplies by nine. 7. No. Check probability assumptions, changing conditions, data quality and implementation errors as well.
## 10. Completion check
Submit the two path tables, their maximum drawdowns and a paragraph distinguishing random variation from model error. A useful conclusion states what the model assumes and what the observed sequence can and cannot show. It does not promise eventual recovery or use a losing run as a reason to increase stakes.
Review [probability](https://betting.crazywingo.ph/article/implied-probability-bookmaker-margins-beginners), [returns](https://betting.crazywingo.ph/article/read-betting-odds-calculate-payouts), [budget boundaries](https://betting.crazywingo.ph/article/betting-budget-limits-avoid-chasing-losses), [market types](https://betting.crazywingo.ph/article/moneyline-spread-total-parlay-beginners) and [market identification](https://betting.crazywingo.ph/article/sports-betting-basics-for-beginners) as needed. The next lesson examines sample sizes and impressive-looking win rates.