Combined Results

Understanding Combined Totals

Before applying this to betting, it helps to understand why a shared result behaves differently to the "and" and "or" situations. With those, you are always tracking separate yes/no outcomes. A combined total asks something different, whether a number made up of two contributions reaches a threshold.

Start with two dice

Imagine rolling 2 dice and asking whether the total of both adds up to 8 or more. This isn't the same question as "did the first die show a high number" or "did the second die show a high number" individually. A low roll on one die can still clear the threshold if the other die rolls high enough to make up for it. A 3 and a 6 together clears it. So does a 5 and a 5, or a 6 and a 4.

To work out the true chance of the total reaching 8 or more, you'd need to go through every possible combination of the two dice and count up which ones add to 8 or more against all the combinations possible. That's a very different task to just multiplying or adding two individual chances together, since the two contributions can trade off against each other in dozens of different ways.

The same logic as X out of N, just extended

This "list every combination" approach isn't new. Take Player 1 and Player 2, each with their own odds to score a try. There are only 4 possible outcomes between them, they both score, only Player 1 scores, only Player 2 scores, or neither scores. To find the chance of at least 1 try between them, you'd list all 4 outcomes and sum up every one where at least one player scores.

The difference with a combined total is what each contributor can produce. Player 1 and Player 2 scoring a try is binary, each either did or didn't. A combined total swaps that binary in/out for a range, each player could contribute 0, 1, 2, or more of whatever's being counted. The list of possible outcomes gets a lot longer, since you're no longer combining two yes/no answers, you're combining two ranges of values. The underlying idea, list every combination, sum the ones that qualify, stays exactly the same.

Why this is harder to pin down

With "and" and "or," you only ever needed one number per person or team, the chance of their own outcome happening. With a combined result, the two contributions blend together, so a strong showing from one side of the pairing can cover for a weak showing from the other. Working out the exact combined probability means accounting for every way the two amounts could add up, not just each side's own chance in isolation.

The two things you always need

Whatever the situation, working out a combined total needs:

• Each contributor's own individual spread of likely outcomes, not just a single chance of success or failure
• The full set of combinations between them that would clear the threshold

Because that full combination approach is rarely practical to calculate by hand, depending on the offer, you may need to use both the Combined Odds calculator for this (read its lesson here) and use the X out of N calculator (its associated lesson here.)