A losing streak in betting is a run of losing bets in a row. From any given bet, the chance of n losses in a row is (1 - p) to the power n, where p is each bet's chance of winning. At $4.00, where a fair chance is 25%, 10 losses in a row from a given start have a chance of 0.75 to the power 10 = 5.6%, about 1 in 18.
Over a season, the useful question is how long the longest run among hundreds of bets will be, and whether your stakes can take it. At $4.00, the longest losing run to expect in 1,000 bets is about 21.
The formula for a losing run
Start with each bet's chance of winning, p. At a fair price p = 1 / odds; a bookmaker's margin makes the true chance a little lower, and a real edge makes it a little higher. Then:
chance of n losses in a row from a given start = (1 - p) to the power n
With p = 1 / price (illustrative, no margin and no edge):
| Price | Win chance | 5 losses in a row | 10 in a row | 20 in a row |
|---|---|---|---|---|
| $2.00 | 50% | 3.1% | 0.10% | About 1 in a million |
| $4.00 | 25% | 23.7% | 5.6% | 0.32% |
| $8.00 | 12.5% | 51.3% | 26.3% | 6.9% |
| $15.00 | 6.7% | 70.8% | 50.2% | 25.2% |
At $15.00, 10 straight losers from a given start is close to a coin toss: (14 / 15) to the power 10 = 50.2%.
Mixed prices and the margin
When the prices differ, multiply each bet's chance of losing. With illustrative prices and each chance of winning taken as 1 / price, the chance of losing at $2.50 is 1 - 1 / 2.50 = 0.60. Six losers in a row at $2.50, $4.00, $3.00, $6.00, $2.00 and $5.00 then have a chance of 0.60 x 0.75 x 0.667 x 0.833 x 0.50 x 0.80 = 0.10, or 10%.
Small changes in the true chance matter. At $4.00, if the true chance is 23% rather than 25%, 10 in a row from a given start becomes 0.77 to the power 10 = 7.3%; at 27% it is 0.73 to the power 10 = 4.3%.
Bets that are not independent
The formula assumes each result has nothing to do with the last. Two runners backed in the same race cannot both win, so count the race, not the bets; a multi is one bet with one chance, however many legs it has; and bets on the same game win and lose together. Counting linked bets as separate makes the formula give the wrong answer.
The chance of a long run somewhere in your bets
The 5.6% is the chance that a run of 10 starts at one chosen bet. Over hundreds of bets there are hundreds of places for a run to start, so the chance that one turns up somewhere is far higher. At $4.00, counted exactly:
| Run of at least | In 100 bets | In 250 bets | In 500 bets | In 1,000 bets |
|---|---|---|---|---|
| 10 losses | 79% | 98% | Over 99.9% | Over 99.9% |
| 15 losses | 27% | 57% | 82% | 97% |
| 20 losses | 6.5% | 17% | 32% | 55% |
| 25 losses | 1.5% | 4.2% | 8.7% | 17% |
So in 1,000 bets at $4.00, a run of 20 losers is slightly more likely than not, even though a run of 20 starting at any one bet is a 0.32% chance.
The exact count keeps one figure for each current run length from 0 to n - 1. Each bet moves a run up by one with chance (1 - p) or back to 0 with chance p, and the share that reaches n is the answer. In a spreadsheet that is one row per bet and one column per run length, plus one for the share that has reached n: 21 columns for a run of 20.
A shortcut often quoted for this, 1 - (1 - (1 - p) to the power n) to the power (bets - n + 1), counts overlapping stretches as separate tries. It overstates the chance badly: for 20 losers in 1,000 bets at $4.00 it gives 95.6%, against the exact 55%.
The longest losing run to expect
As an approximation, the average longest losing run in a set of bets is:
approximate longest losing run = (ln(bets x p) + 0.58) / ln(1 / (1 - p)) - 0.5
where ln is the natural log on any phone calculator. For 1,000 bets at $4.00: ln(1,000 x 0.25) = ln(250) = 5.52, plus 0.58 is 6.10; ln(1 / 0.75) = 0.288; 6.10 / 0.288 = 21.2, and minus 0.5 that is about 20.7. With p = 1 / price:
| Price | Win chance | Longest run to expect in 250 bets | In 1,000 bets |
|---|---|---|---|
| $1.90 | 52.6% | 6.8 | 8.7 |
| $4.00 | 25% | 15.9 | 20.7 |
| $8.00 | 12.5% | 29.6 | 40.0 |
| $15.00 | 6.7% | 48.7 | 68.8 |
For these cases the approximation lands within about a tenth of a loss of the exact average. It is an average, not a ceiling: the longest run in any one season varies around it, and at $4.00 there is about a 17% chance of 25 or more in 1,000 bets.
Runs grow with the logarithm of the number of bets, so four times as many bets adds only about 2 losses at $1.90 but about 20 at $15.00 (48.7 to 68.8). The price matters far more than the count: someone backing $8.00 chances should expect runs more than four times as long as someone betting line markets at $1.90.
Planning stakes around long runs
A plan that survives a losing run is set before the run starts. Four steps:
- Note your usual price and how many bets you place in a year.
- Work out the longest run to expect, and a longer one you would see only about 1 year in 20.
- Work out what each costs at your stake.
- Pick a stake at which those costs stay inside a share of the bankroll you decided in advance you can afford to lose.
With illustrative numbers: bets at about $5.00, about 1,000 a year. The longest run to expect is (ln(200) + 0.58) / ln(1.25) - 0.5 = 25.8, about 26. Counted exactly, a run of 37 or more turns up in about 1 year in 20 (4.9%).
| Stake | Level stakes, 26 losses | Level stakes, 37 losses | Share of current bank, 26 losses | Share of current bank, 37 losses |
|---|---|---|---|---|
| 0.5% | 13% of the bank | 18.5% | 12.2% | 16.9% |
| 1% | 26% | 37% | 23.0% | 31.1% |
| 2% | 52% | 74% | 40.9% | 52.6% |
| 3% | 78% | 111%, more than the bank | 54.7% | 67.6% |
Level stakes cost the run length x the stake: 37 x 1% = 37%. Staking a share of the current bank costs 1 - (1 - stake) to the power of the run length: 1 - 0.99 to the power 37 = 31.1%, a little less because each stake shrinks with the bank.
A straight run is also only part of a bad patch, and the full fall from a high point is usually deeper, which variance in betting measures. How stake size moves the odds of going broke altogether is the subject of risk of ruin.
Put a deposit limit on every betting account at a level your plan can carry. Australian online bookmakers must offer one under the National Consumer Protection Framework; a cut takes effect at once, while a rise waits 7 days.
Risk: Betting involves risk. A stake sized for the expected run still loses money in a longer one, and these runs assume each bet's chance is exactly 1 / price. There is no guarantee of profit at any stake. See responsible gambling for limits and support.
What a losing run says about your method
Not much on its own. Check the run against the tables for your usual price before you change anything: at $8.00, 30 losers from a given start is a 1.8% chance, yet the longest run to expect in 1,000 bets is about 40.
If the run sits inside what the price makes likely and your bets keep beating the closing price, it is most likely luck. If it sits far outside, or your prices have stopped beating the close, look at the method rather than the run. Judging a method by its prices instead of its last 20 results is the core of a sound betting strategy.
Never chase a losing run
A losing run says nothing about the next bet. After 10 losses at $4.00 the next bet still wins about 25% of the time, and believing a winner is due is the gambler's fallacy.
Raising stakes to win back what a run cost only makes the run more expensive. On the $5.00 plan at 1% level stakes, a 26-loss run costs 26% of the bank. Double the stake halfway, after 13 losses, and the same run costs 13 x 1% + 13 x 2% = 39%. Doubling after every loss is the martingale system, and long runs like these are exactly where it breaks.
If you notice yourself betting more often, for longer or with more money than you planned to get a run back, stop there: chasing losses covers what to do instead. For free and confidential support call 1800 858 858 or visit gamblinghelponline.org.au.
Risk: Betting involves risk. Every run on this page assumes independent bets at the stated chance, real chances can sit below what the price implies, and no stake size makes a long losing run impossible. See responsible gambling for limits and support.