# Two random sets, how to formulate what percent of set A is bigger than set B.

Okay, so this is almost a math question as much as a computer programming question. I'm thinking the math should be easy, but setting the formula is tripping me up.

I have an excel spreadsheet of football teams/stats/scores, etc. In the second sheet, I lay each matchup for the week and the sheet lays out the Upper Quartile and Lower Quartile that each team could probably score. Add in the current casino line (yes, its probability team A beats B after points are given/taken), and the spreadsheet will give results such as:

Team - UQ - LQ A..... -32-20 B..... -35-23 Betting line, team A, +3

Now, assuming 50% of each teams score fits within its Upper to Lower quartiles, looking at this example, the result would be 50% probability (I'm not great at statistics, but it seems so to me) that team A wins, 50% team B wins (given that the range of quartiles after betting line is added becomes equal). The basic question is: how does one mathematically get that number? There are scenarios where the math I've come up with gets all messed up, I'll example some below:

A... 21-16 B... 19-13 Line +4

This should return 100%, given that after the line is added, 100% of team A's values are above B's values

A... 27-17 B... 31-12 Line -3

Now we subtract the line from team A's score and we get some outputs where B will always win/never win/might win. This would seem a less than 50% chance a random score from team A will be higher than a random score from team B. How would all these be calculated? Is there an easy enough way, given this information, to write a one line math formula to figure it out?

Sorry I can be long winded-but thank you for getting this far, all help is appreciated.

• Your title suggests a problem (how much "of set A is bigger than set B?") that really isn't well-defined mathematically. While there are special cases (all of set A is bigger than anything in set B) that one could make a persuasive answer for, in general it doesn't have a clear sense. Perhaps you should think in terms of probability from the outset. Suppose a probability distribution is defined on the product of two sets, $A\times B$, with the interpretation that a pair of scores is being sampled for the two respective teams. One then has a well-defined chance for one team to win. – hardmath Sep 26 '18 at 5:09
• hardmath-Yes, I know the headline isn't as clear as I can be, its been a hard question for me to specify/quantify. I am trying to think of probability, yes, but given the distribution should be normal (an equal chance of any number in that range, isn't the most precise given football favors denoms of 3 & 7, but I'm not trying to be excessively technical), so just say normal distribution of range A and B, I guess the basic question is "how to calculate a formula/probability where range A (numset) will randomly produce a higher number than range B" Ranges will be defined, but different values. – Benjamin Sisley Sep 26 '18 at 6:01
• As a mathematical modelling problem you can decide how you want to assign the probabilities for the outcomes in $A\times B$. While $3$ and $7$ scores are favored, a safety contributes $2$ points and makes all scores except $1$ possible. Realistically the score in $A$ is not independent of the score in $B$, since time of possession (and scoring opportunities) will tend to favor one team over the other. But mathematically, once you've assigned probabilities to the joint distribution on $A\times B$, you've reduced the computation of the winning chances to a formality. – hardmath Sep 26 '18 at 6:17
• hardmath, I know the computation becomes a "formality".. because, well, once you add in the line, then trying to sort out the more probable and less probable nums gets a bit harder.. I'll add a picture below of what I'm working.. – Benjamin Sisley Sep 26 '18 at 23:16
• Okay, of course taking a photo to show you isn't working... nonetheless, lets take tomorrows game; Vikings/Rams.. I have a range for MIN of 11-22points, and LAR of 27-31.. The line for MIN is +7.. So if MIN is going to score between 18-29pts and LAR is going to score 27-31.. what percentage chance does MIN have of covering the spread? What formula would calculate that? – Benjamin Sisley Sep 26 '18 at 23:27