Correlated bets are bets whose outcomes are linked, so one landing changes the chance of the other, and the chance of both is not their two chances multiplied. When the link is positive the pair lands together more often than multiplying suggests; when it is negative, less often.
That is why a bookmaker will not let related legs into a standard multi at multiplied prices, and why a same game multi gets one price for the whole set. Multiplying is right only for independent outcomes, the case how betting odds work covers for an ordinary multi.
Positive and negative correlation, with sports examples
Two outcomes are positively correlated when one makes the other more likely, and negatively correlated when one makes the other less likely. With no link at all they are independent, and only then does the chance of both equal the two chances multiplied. Some pairs, with the direction of the link:
| Pair | Direction | Why |
|---|---|---|
| A tennis favourite to win, and the match under its total games line | Positive | A straight-sets win keeps the game count down |
| The same favourite to win, and the match over that line | Negative | A long match usually means the underdog took a set |
| A soccer team to win, and the same team to keep a clean sheet | Positive | Conceding nothing makes a win far likelier |
| A T20 team batting first to make 200 or more, and that team to win | Positive | A big total is hard to chase |
| A basketball team to win, and the opposing team's star to score 40 or more points | Negative | A huge night from the star usually lifts the other side |
| Runner 1 to win a race, and Runner 2 to win the same race | Negative, almost completely | Only one runner can win outright; a dead heat for first is settled under the bookmaker's dead-heat terms |
| A team to win a knockout final, and the same team to win the premiership | Positive | Losing the knockout final ends its premiership chance |
The last row shows that legs from different matches can still be linked. The strength of each link is what a price has to capture, and it differs from game to game.
Two legs that move together, worked
Take an illustrative tennis match with fair chances, the margin already out. Player A, the favourite, wins 70% of the time, and the match goes under 21.5 total games 50% of the time. Here is how the two outcomes share out across every result:
| Under 21.5 games | Over 21.5 games | Total | |
|---|---|---|---|
| Player A wins | 42% | 28% | 70% |
| Player B wins | 8% | 22% | 30% |
| Total | 50% | 50% | 100% |
Multiplying the two chances says A wins with the under 0.70 x 0.50 = 35% of the time, a fair price of 1 / 0.35 = $2.86. The table says 42%, because A's wins tend to be short matches, so the pair's fair price is 1 / 0.42 = $2.38. The other pair runs the opposite way: A with the over lands 28% of the time, a fair price of 1 / 0.28 = $3.57, not $2.86.
Now an illustrative bookmaker: Player A $1.36 and Player B $3.10 (a 1 / 1.36 + 1 / 3.10 = 105.8% market), under and over 21.5 games $1.87 each (2 / 1.87 = 107.0%). Multiplied, either pair is 1.36 x 1.87 = $2.54:
| Pair | Fair price | Legs multiplied | Average return per $1 at the multiplied price |
|---|---|---|---|
| A wins, under 21.5 | $2.38 | $2.54 | 0.42 x 2.5432 = 1.068, about 6.8% above fair |
| A wins, over 21.5 | $3.57 | $2.54 | 0.28 x 2.5432 = 0.712, about 28.8% below fair |
Even with a margin in every leg, the positive pair multiplied is worth more than it costs. The negative pair multiplied is a very poor bet.
Risk: Betting involves risk. These chances are illustrative, a bookmaker can refuse a related combination or settle it under its own terms, and even the better-priced pair misses 58% of the time. See responsible gambling for limits and support.
Why bookmakers block or reprice related contingencies
A related contingency is the betting term for correlated multi legs, where the result of one changes the chance of another. The worked example shows the problem for a bookmaker. A standard multi multiplies its legs, and on positively related legs that can pay more than the combination is worth: by 6.8% in the tennis pair, even after each leg's margin.
So bookmakers handle related legs in one of two ways:
- Refuse the combination in a standard multi, so the bet slip will not accept the second leg.
- Price the combination as one bet, with its own price set from their view of how the legs move together, plus a margin. That is what a same game multi or a same race multi is.
Which legs a bookmaker treats as related, and which markets it lets you combine, are set in its own terms. So is what happens if a related multi gets through, so check them before you rely on a price. For NSW-authorised sports bookmakers, the Bookmaker Declared Betting Events Betting Rules allow all-up bets across different events (clause 4.2) and leave it to the bookmaker which outcomes and contingencies it takes bets on (clause 4.3.1).
Risk: Betting involves risk. A related multi that gets through can still be voided or resettled under the bookmaker's terms, so the 6.8% may never be paid, and even that pair loses more often than it wins. See responsible gambling for limits and support.
Same game and same race multis are priced correlation
A same game multi (SGM) is a bookmaker's own price for a correlated combination. Say an illustrative bookmaker offers A with the under at $2.20 and A with the over at $3.30. Against the multiplied $2.54, it has cut the first by 2.20 / 2.5432 = 0.865, about 13.5%, and lifted the second by 3.30 / 2.5432 = 1.298, about 29.8%. On the fair chances both return the same: 0.42 x 2.20 = 0.924 and 0.28 x 3.30 = 0.924, an average loss of 7.6 cents per dollar either way.
So an SGM below its legs multiplied is not necessarily poor value, and one above them is not necessarily generous. Each has to be judged against the chance of both legs landing. How same game multi odds are calculated takes an SGM price apart, and same game multi strategy covers building one.
A same race multi does the same for legs in one race. Legs on different runners usually pull against each other, because they compete for the same places, and legs on the same runner overlap. Same race multi works an SRM price against its legs multiplied.
How to measure correlation in betting
Three numbers describe a link, all from the table above:
- The conditional chance, the chance of one leg given the other lands. The under lands in 0.42 / 0.70 = 60% of A's wins, against 50% across all results, and in only 0.08 / 0.30 = 26.7% of B's.
- The lift, which is the joint chance divided by the two chances multiplied: 0.42 / 0.35 = 1.20 for A with the under and 0.28 / 0.35 = 0.80 for A with the over. A lift of 1 means no link. Divide the multiplied fair price by the lift to get the pair's fair price: 2.857 / 1.20 = $2.38.
- The correlation coefficient, which runs from -1 to +1. For a two-way table it is (0.42 x 0.22 - 0.28 x 0.08) / square root of (0.70 x 0.30 x 0.50 x 0.50) = 0.07 / 0.229 = +0.31 for A with the under.
The coefficient is the statistician's number, but the lift is the punter's: it turns straight into a price.
Spotting correlation the price does not reflect
A price can only build in the link the bookmaker sees. To test one:
- Take each leg's fair chance from a market with the margin removed.
- Ask what one leg landing does to the other, and estimate that conditional chance from how the game would have to go, or from results of similar games.
- Joint chance = first leg's chance x the second leg's chance given the first. The fair price is 1 / joint chance.
- Compare: average return per $1 = joint chance x the price on offer.
- Check the same combination at other bookmakers, because each prices the link its own way.
Links are easiest to miss where they cross markets: two props on the same player, a futures leg beside a match leg, or conditions on the day that move several markets at once. A pricing model built for typical games can also miss what is unusual about this one.
Risk: Betting involves risk. A conditional chance is the hardest number in betting to estimate, a price that looks generous may reflect a rule you have not read, and a well-priced pair still misses most of the time. There is no guarantee of profit. See responsible gambling for limits and support.
Mistakes with correlated legs
With the tennis numbers above:
| Mistake | What it costs | The fix |
|---|---|---|
| Judging an SGM against its legs multiplied | The $2.20 SGM looks 13.5% below the multiplied $2.54, but its real cost is 7.6% against the fair $2.38 | Compare with 1 / the joint chance |
| Taking a negative pair above its multiplied price as a bargain | A with the over at $3.30 is 29.8% above $2.54 and still 7.6% below the fair $3.57 | The same check: 0.28 x 3.30 = 0.924 |
| Adding a leg that sits inside another | Over 21.5 games and over 20.5 games: the second leg adds nothing but its margin | Keep only the leg that carries your view |
| Treating legs from one competition as independent | A knockout final and the premiership move together, so their chances do not multiply | Ask what one result does to the other |
Adding legs to a multi of any kind has its own cost, and multis vs singles works it through.
Leg prices for a correlation check on B337's Terminal
Pricing a link starts with each leg's own fair chance. The Terminal shows sports prices side by side on head to head, line and total markets, and player props where bookmakers offer them. The best available price on each selection is picked out, with exchange back and lay prices beside it. Across racing and sports it covers 40+ bookmakers. Because prices are refreshed on a repeating cycle, a price on screen can lag the bookmaker's own, so check it in your bookmaker account before you bet. A free account opens a limited view of the Terminal with live odds.
Risk: Betting involves risk. A fair chance is an estimate, linked legs lose together as well as win together, and a bookmaker can restrict an account or void a bet. See responsible gambling for limits and support.