Variance in Betting: How Likely Losing Streaks and Bankroll Drawdowns Are

Variance in betting is the spread of results around the expected value. Even if your approach makes a profit on average, over a short stretch the result is decided by chance: there will be winning streaks, losing streaks and months in the red. A long losing streak is not a sign of a mistake but an ordinary property of randomness, and its probability can be worked out in advance.

One thing up front: betting carries the risk of losing money, and no calculation guarantees a result. You can work out the probability of a streak and the drawdown for your own numbers in the calculator below; it is also available on its own page — the losing streak calculator.

A losing streak in betting: how likely is it

Intuition is bad at judging streaks. Five losses in a row at a 50% chance seem rare: the probability of five specific bets all losing is only about 3%. But there are many bets, and a streak can start on any of them. So the right question is not "what is the probability of five losses in a row" but "what is the probability of at least one such streak over the whole run".

Here is the exact calculation for bets with a win probability of 50% — for example, at odds of about 2.00:

The longest losing streak in 100 such bets averages about six, and in 500 bets more than eight. A streak of five losses in a hundred bets happens in about four cases out of five, and a streak of six in about every second case.

Streaks depend heavily on the win probability. If you back outsiders with a probability of about 40%, the longest streak in 100 bets averages almost eight losses, and a streak of eight happens in about half of cases. Even on favourites with a probability of 60%, a streak of six losses in a row within 500 bets happens in about 71% of cases.

A bad run in betting and bankroll drawdown

A drawdown is the fall of the bank from its peak. How much a streak takes depends on the stake size and on how it is worked out.

Flat staking — every stake is a percentage of the starting bank. A streak of L losses takes L such stakes. With a stake of 5% of the bank, six losses in a row is minus 30%; with 2%, minus 12%. A streak of ten at 2% costs a fifth of the bank.

A percentage of the current bank — every stake is worked out from what is left. After a loss the stake shrinks, so the bank falls more slowly: six losses at 5% of the current bank is minus 26.5%, not 30%. With this approach the bank never reaches zero, although it can become very small.

Hence the practical rule: choose the stake size not by how much you want to win but by what streak the bank has to survive. If a streak of eight to ten losses is likely over your number of bets, a stake of 10% of the bank turns a bad run into a disaster, and a stake of 1–2% into an unpleasant but survivable episode. More on choosing a stake size and setting limits is in the article on bankroll management.

Bankroll drawdown happens even with a real edge

The nastiest property of variance is that it works on everyone alike. Take bets at odds of 2.00 with a real win probability of 52.5% — an edge of +5% per bet, which is a lot in betting. Even with that edge the player is in the red:

• after 100 bets — in about 27% of cases;
• after 500 bets — in about 12% of cases;
• after 1,000 bets — in about 5% of cases.

And a player with no edge at all is up after 100 bets in almost half of cases. That is why a hundred bets say almost nothing about the quality of an approach: a lucky run looks like skill and an unlucky one like a mistake. An approach can only be judged on a large sample and on how well your probability estimates match reality — more on this in the article on how to measure prediction accuracy.

How to survive a losing streak

1. A small stake. 1–2% of the bank per bet is the main protection. The calculator in this article shows how much of the bank is left after a streak that is likely over your number of bets.
2. Do not chase losses. Raising the stake after losses is a progression, and it makes the drawdown deeper. Why chasing does not work is covered in the article on Martingale and other progressions.
3. A log instead of memory. Record every bet: after a few hundred bets you will see where variance ends and a systematic error begins.
4. A stop set in advance. Decide before the season at what drawdown you take a break and review your approach.
5. Comparison with an independent estimate. If your probability regularly differs from the market and from a model, that is a reason to check it. Estimated probabilities for upcoming matches are in football predictions.

The short version

Variance in betting makes long losing streaks routine: at a 50% chance, a streak of five or six losses in a hundred bets is more likely than not. The drawdown is set by the stake size, so choose it for the streak the bank has to survive. And even a real edge does not guarantee a profit over a few hundred bets.