Something I've noticed a lot on TV is hosts/announcers using statistics incorrectly by converting past performances into present odds. For instance:
A football announcer says a kicker has "a 16% chance" of making a kick from the 61-yard line.
A host in a battle royale competition says each of the last four competitors has "a 25% chance" of being the winner.
This clearly cannot be correct from the point of view of statistics. In the case of kicker, surely the announcer means that of all the kicks from the 61-yard line, only 16% have been good -- which is relevant to the current situtation but there are clearly other factors involved. In the case of the battle royale, one person has a 100% chance, and the other three have a 0% chance.
This is a post about math and not word choice though so my question is this: Is there a specific way to compute/represent what the odds are at that given moment? I'm thinking through a Bayesian representation of the most relevant factors involved.
In the case of the kicker, I figure there's the ability of the individual kicker to factor in as well as factors like wind speeds, the abilities of the snapper and holder, home vs. away, etc. It seems it would be a similar analysis for the battle royale although obviously, the relevant factors under consideration will be different.
How would such a factor be computed? By simply multiplying the odds of each contributing factor together? 1-the Baysean times everything else? Or something else I'm not considering?
I get this change is unlikely to happen, but if it could, and did, what would it look like, as math?
You ask what might [a rigorous statistical analysis] look like? Well you'd collect a lot of past data and then make a judgment call to decide to the criteria you are going to base your analysis on. E.g., in the football example, you might decide to make your criteria be who the kicker is (or which team he plays for, or who coached him or ...) and which lines the kick was from. You then have a well-defined statistical problem. How valid the results of the statistical analysis are depends on whether you chose a good set of criteria: something you could refine over time.
Another example is weather forecasting. I believe the meteorologists come up with their percentage probabilities of rain say, by using past data to create a statistical model of how rain-bearing clouds spread or shrink as they move through the atmosphere. So based on present data about cloud coverage and wind speeds they can estimate for a given area of the earth the percentage of that area that will be covered by rain-bearing clouds at some time in the future. The forecast for my local area for tomorrow at 14:00 is 63%, meaning that their model predicts that 63% of the area will be covered by rain-bearing clouds. I.e., if you pick a random point in the area there is a 63% chance that it will be raining at that point. These predictions (at least in the UK) tend to be pretty accurate these days (and you can see what's going on on the day via weather radar) - so I imagine that there has been a great deal of refinement and that many more complex criteria than the ones I have mentioned are taken into account.