Intro to Probability for Fantasy Football

Here’s a quick set of numbers –

17... 24... 8... 4... 13... 12... 0... 15... 14... 13... 11... 7... 0... 8... 17... 23... 2... 15... 9... 4... 28... 5... 9... 5... 17...

No, not a winning lottery ticket. It’s 25 randomly selected weekly RB performances from last year. The first is Todd Gurley's 17 point performance in week 16, the second is Derrick Henry's 24 point performance week 14, etc.

Ok. Now let’s arrange these from smallest to largest, marking each with an

and stacking

‘s when a score shows up multiple times. Make sense? That gives us something like this:

This is interesting, but why should we limit ourselves to just 25 games? Let’s see what it looks like when we plot ALL the RB’s over ALL their games:

Here’s the crux of thinking probabilistically about fantasy football:

. Each

is equally likely to get picked. Each

, however, is not. There are a lot more

‘s between 0-10 points than there are between 20 and 30.

Now let’s make a quick shift from

's to curves, which are easier to deal with mathematically. The general shape above can be represented by something like this, which we’ll call a

:

So we’re heading into the Monday night game and you’re up by 10 points. Your team is done and the guy you’re playing only has one RB left — how likely is it you come out with a win? The answer — assuming you don’t know anything about the other RB — is about 50/50. Half the area of the curve (half of the little x's) is at 11 points or higher, half is below.

What if you do know something about the other RB — say, that it’s Ezekiel Elliot? In that case your 50/50 odds drop precipitously, because Zeke is not some random RB. His score is, on average, higher than a random RB, which means his distribution is further to the right, something like the red one below.

This is important.

. A lot of information goes into it — offensive line, this week's opponent, whether he’s in a committee, and so on. In this case Zeke will score more than 10 points 91% of the time and things aren’t looking good for Monday night.

Another thing that's important: the differences in a lot of these curves is very small. Here are the top 20 RBs for week 14 last year. Being furthest to the right, CMC is the "best", and will be the highest scoring RB more often than anyone else. But there's an awful lot of overlap.

Come Sunday at noon we're drawing one x each from 20 players. The actual observed outcomes could realistically be anything.

These close, overlapping distributions is what people mean when they point out how much luck is involved fantasy. It also makes going back and making sense of what just happened incredibly difficult.