# Algorithm to find the probability of a number given the probability of another number?

Let me preface that I'm not sure if this is even possible and I'm unsure of what math principals would apply but I'm trying to build an algorithm for a sports betting project that can find the probability of a number given the following assumptions:

• A QB has a 53% chance of going over 282.5 passing yards (I'm getting this line and odds from the sports book's implied odds)

• In their previous games they had the following passing yards:

[262, 308, 282, 309, 249, 466]

• The team they are playing has given up the following passing yards to QBs in previous games:

[316, 364, 146, 290, 222, 186]


Are there any math principles that could estimate the probability of the QB going over 249.5 yards?

I've thought maybe calculating the z score for the QB's previous yards might be it:

z = (x − μ) / σ

• μ = 312.67
• σ = 72
• z = (249.5 - 312.67) / 72
• P(x>Z) = 0.81 or ~81% chance of going over

But that doesn't really factor in the assumption that they have 53% chance of going over 282.5 yards or the opponent's previous yards allowed.

So with that I'm not sure if this is possible, or if more information / data is needed or if there's a completely different math principle used to solve something like this.

• There are many issues lurking here, so let me just ask about the first one that crops up: how do you know that "A QB has a 53% chance of going over 282.5 passing yards"? That kind of assertion appears in some textbooks and exam questions, but it's never true of the real world, so it must be code for some kind of information. What is that information?
– whuber
Commented Oct 17, 2023 at 22:30
• @whuber that's correct, I should've explained in the question, I'll go back and edit. But those are the implied odds that I get from sportsbooks
– AJK
Commented Oct 18, 2023 at 2:53
• Do you have access to more than 6 data points? And do you know how the score of the QB (first vector of data) are related to the score of the QBs in the team (second vector of data)? Commented Oct 19, 2023 at 18:15
• @CamilleGontier I could probably go back further seasons for more data points, although I'm not sure if they would be as relevant since rosters / defenses etc. change year to year. So right now both vectors are just reflecting the current NFL season. The 2nd vector is just based on how the defense he's playing has done vs previous QB's they've gone againse, so I was thinking if they've given up more yards than the league average (I probably need the league average now to factor that in maybe?) then that might factor in to how well the QB might do
– AJK
Commented Oct 19, 2023 at 18:24
• Will you change “principal” to “principle”, and round the numbers to two or three digits after the decimal point? That would make the question more inviting of a serious answer. Commented Oct 23, 2023 at 5:24

Your question is primarily concerned with $$P(Y|A)$$ where Y is allowed yards for your QB of interest and you condition that probability on the opposing team's historic allowed yards.

The comments on your question are correct that there is a lot of unknown variables or model decisions here, but if you are comfortable either assuming things for convenience of calculation or with being representative of the uncertainty, then you can probably still get a model for it with varying certainty.

If you know this QB gets >= 282.5 yards 53% of time unconditional on opposing team (averaged), then a question is what is the distrib of passing yards for a player?

It may be fine to assume Gaussian for Y if they always perform similarly each time independent (enough) of the opposing team and other external factors.

Given the (very small amount of) data, the assumed model, and an accepted 53% prob at that point, you can construct a Gaussian for Y that satisfies the data and probability constraint. Fit a Gaussian whose 53% of cumulative prob is at the 282.5 yards and hold that as true as you find the sample standard deviation given that constraint. The result can serve as Y.

The same Gaussian assumption could then apply to A for the opposing team. Similarly fit it the best you can with the data available and fill in any variables with best guesses or use em as parameters to explore possibilities.

You then may think of that as two opposing forces Y vs A and so you can think of how they would interact to one another. With our assumption of Gaussians, you could take the convolution and get a resulting Gussian that would indicate the probability of passing yards Y given team's allowed yards A. Y * A

Then you could do your z score test and it at least incorporates all the information at the cost of what may be a naive model.