# Intro to Probability for Fantasy Football

Last updated: July 27th, 2020

## How to think about weekly player performances

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

x

and stacking

x

‘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:

for
any given week, when you start a player you’re picking out one of these little
x's at random

. Each

x

is equally likely to get picked. Each

score

,
however, is not. There are a lot more

x

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

Now let’s make a quick shift from

x

'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

distribution

:

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.

The shape of this curve changes for any given player and
week

. 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.