X and Y to occur
Understanding X And Y To Occur
Before applying this to betting, it helps to understand how the chance of two separate things both happening compares to the chance of either one happening alone.
Start with two independent things
Imagine you flip 2 coins. You want to know, what's the chance that both land heads?
Each coin on its own has a 50% chance of landing heads. To find the chance of both landing heads, you multiply the two chances together. 50% x 50% gives you 25%. That's lower than either coin's individual chance, which makes sense, since asking for two things to happen is always harder than asking for just one.
This is the core rule for combining two things with "and" by multiplying their individual chances together, as long as the two things don't affect each other.
When the two things aren't independent
The coin example works cleanly because one flip has no effect on the other. But plenty of real situations involve two things that move together rather than independently.
Take a player scoring 2 goals and their own team winning the match. These aren't independent. If a player scores twice, it's quite unlikely their team goes on to lose, so the two outcomes are positively correlated, they tend to happen together more often than pure independence would suggest.
Flip it around and the opposite shows up. Take the same player scoring 2 goals, but this time paired with the opposite team winning. That's negatively correlated, since it's unlikely a player scores twice and their team still loses.
Why this matters for the maths
Multiplying the two individual chances together only gives you the right answer when the two things are truly independent. Once a connection like the above exists, that simple multiplication drifts away from the real answer, understating the true chance when the correlation is positive, and overstating it when the correlation is negative.
The two things you always need
Whatever the situation, working out the combined chance of "X and Y" needs:
- The individual chance of each thing happening on its own
- Whether the two things are independent, positively correlated, or negatively correlated
Seeing correlation as a real number
Take Sharks vs Knights in the NRL. Bet365 offers $2.30 for Dominic Young to score a try anytime, $1.30 for the Sharks to win, and $3.20 for the Knights to win.
Multiplying Young to score by Knights to win gives 2.30 x 3.20 = 7.36 if the two outcomes were independent, that's the price you'd expect for both happening together.
But Bet365 actually prices Young to score and Knights to win at $4.75, well under that 7.36 figure. Dividing the actual price by the independent price (4.75 รท 7.36) gives a correlation factor of about 0.65. A factor below 1 confirms the positive correlation, Young scoring makes a Knights win more likely, so the combined price comes in lower than simple multiplication would suggest.
Now flip it to Young to score and Sharks to win. Bet365 prices that combination at $3.40, doing the maths of 2.30 x 1.30 = 2.99, this time our straight multiplication is below the offered odds. Dividing 3.4 by 2.99 gives a correlation factor of about 1.14. A factor above 1 confirms the negative correlation, Young scoring makes a Sharks win less likely, although only slightly as the Sharks are a strong favourite, so the combined price comes in higher than simple multiplication would suggest.
To estimate a correlation factor like this yourself, build the same game multi across a few different bookies and compare the combined price each one offers against the independent multiplication. Not every bookie applies the same correlation. One might treat a pairing as strongly connected while another treats it as barely connected at all, and in some cases, like two players from different teams to each score a try, one bookie might apply positive correlation while another applies negative. Checking more than one book is the only way to catch that disagreement, and once you have a few figures, using the one that gives the least favourable price is generally a safe number to use, allowing us the err on the side of caution when calculating. However it does risk over estimating in the wrong direction, so another aproach is to take a group of these and use the average for you calculation, taking into account groups that use the same odds aren't really seperate selections.