Rookie Running Backs and a Famous Statistics Problem

[jeffster]

This is a fun exercise in basic probability, with some applications to football. If that doesn't sound fun to you, please skip it!

In his first appearance as a pro, Ronald Jones ran for 9 yards on 8 carries. Reports from camp suggest he's struggling with pass protection and receiving. Now obviously it's far too early to pass judgment on any of the rookies, but it did get me thinking of a particularly famous problem in statistics. If you're paying attention to the numbers in this game then it's important that we avoid drawing conclusions they don't support, so let me get hypothetical and explain!

There are a bevy of backs that share a tier in the first round this year. The most recent DLF ADP has Guice, Penny, Michel, Jones, Chubb, Freeman and Johnson all ranked between 1.02 and 1.09, and I think we all know that some of these guys will bust. There has been a lot of research into the "hit" rate of 1st round draft picks: Rotoviz suggests 30%-47%, for example, depending on if you exclude guys with one year of success, like Trent Richardson. So here's an example of the question I want to look at:

If I drafted Chubb, and now Jones is a bust, does it raise the odds that Chubb is not a bust?

This is assuming I know nothing new about Chubb (imagine only Jones has played so far), actually do have these RBs in a similar tier (that is, a similar probability of success), didn't draft both of them, or any other complicating factor. On the surface it sure seems like I should be happy Jones is a bust, doesn't it? Only 30% of them are successful on average, and that's one down! I'll cut to the punch line, and then explain: Jones being a bust does not in any way raise the odds that Chubb is not a bust. Counterintuitive, isn't it?

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In statistics and probability, this is called the Monty Hall Problem. It is named for an old game show, and the version we're considering goes like this: a contestant is shown three doors, then told that behind two are goats and behind one is a new car. It is hopefully clear that their odds of getting a new car are 1/3 (33%), which just happens to be pretty similar to the 30% hit rate on first round draft picks. Now for the twist - the host opens one of the doors the contestant didn't pick, revealing a goat. The host then points at the two remaining doors and asks if you want to stick with your original guess, or change to the other door.

Should you change?

Let's get the punch line out of the way again: shockingly, the answer is unequivocally yes, you should change. If you stick with your original pick you only have a 33% chance of a car, but if you switch you have a 66% chance. What? How is it possible I get even better odds than 50% if I switch?! Two doors left, shouldn't it be 50/50 on my second choice? You can prove this yourself by setting up a version of the game and trying it over and over if you like, but I'll try to explain.

It comes down to the information you had when you made the choice. Initially all doors looked the same, thus an even 33% chance for each. Crucially, we know that the host was never going to open our door, so we get no new information about it. Its probability of being a car is fixed at 33%. The other two doors each had their own 33% chance of being the car - in other words, there was a 66% chance the car wasn't behind our door. When the host reveals the goat we didn't pick, recalling three sentences ago when we agreed we got no new information about the door we chose, there has to still be a 66% the car is not behind it. Therefore, the one remaining door now has a 66% chance of being the car.

There are dozens of explanations of this online, though this one is nice because it includes a simulator.

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So bringing it back to football, learning that Jones is a bust gives us no new information about Chubb being a bust, even if we accept a fairly consistent bust rate amongst RBs in this tier. But here's a brain twister for you, now that you know about the Monte Hall problem: if you had Jones, Chubb and at least one other of these RBs in the same tier, and you own Chubb, and Jones is a bust, should you now trade Chubb for the other guy?

Have fun with that!

[collbey]

You bring up an interesting analysis. In the Monty Hall there is no randomness to the door that is showed, you know it will always be a door you didn't choose that is not a new car since someone is making that decision who knows what is behind each door. In fantasy, there is as much chance our guy would be the first bust since no one with inside knowledge is saying who will be the first bust.

[MEuRaH]

There has been a lot of research into the "hit" rate of 1st round draft picks: Rotoviz suggests 30%-47%, for example, depending on if you exclude guys with one year of success, like Trent Richardson.

I wrote a fairly large article talking about hit rates here on these forums a few years ago, and used those rates to make a trade chart. All that info can be found in my sig.

The "hit rate" also is not the same for all 1st round picks. Attached below is a picture that represents the numbers I created/found through research of players that dates all the way back to 1999. If a player maintained a value of multiple 1st round picks throughout their entire career (Jamaal Charles) they would be a "stud" and those boxes are labeled green. If a player maintained a value of a single 1st (Frank Gore) those players were labeled as starters and those boxes are labeled blue. If a player maintained value of something of a role player and they were good enough to be a bye-week fill-in and probably nothing more () they were given a yellow box.

So as you can see below, a player who averaged as a top 3 pick is twice as likely to be a stud than a player picked 4-6. Those numbers drop drastically for picks 7-12.

[BigBawseRoss]

that is just wrong. changing your pick to the other door doesnt increase your odds and the 66% stuff is totally wrong too. once the one door is revealed then both of the remaining doors, including your original door, have a 50% chance of being the right one (as there are now only 2 options, not 3...simple maths)

sorry got lost on the 3 door contest thing haha but i dont really get what the point of this post is. nobody is calling anyone a bust before any of these dudes play at least a season or two so there would never be an opportunity to say oh this guy is a bust but this one might not be

[Dynasty DeLorean]

It doesn’t really work like that. That’s like playing roulette and thinking bc it just landed on red 5 times I. A row now it’s more likely to land on black on the next roll.

[UATahoe]

It doesn’t really work like that. That’s like playing roulette and thinking bc it just landed on red 5 times I. A row now it’s more likely to land on black on the next roll.

Roulette isnt comparable to what is being said here at all. Its a single game of chance every time. What this is relating to is getting to make a second decision based on information from the first. Your first decision basically split the outcome in half but their are 2 parts on the side you didnt pick and 1 part on the side you did pick. Getting to switch doors after 1 has been revealed is allowing you to now choose 2 doors over one when compared to the original decision. The probability is indeed 66.7% or 2/3 if you decide to switch.

The unfortunate part here is you dont get the chance to to run the switch 1,000 times in order for the percentages to even out like you would expect and more than likely get stuck on the 1/3 side with the bag of poop. LOL

[Dynasty DeLorean]

It doesn’t really work like that. That’s like playing roulette and thinking bc it just landed on red 5 times I. A row now it’s more likely to land on black on the next roll.

Roulette isnt comparable to what is being said here at all. Its a single game of chance every time. What this is relating to is getting to make a second decision based on information from the first. Your first decision basically split the outcome in half but their are 2 parts on the side you didnt pick and 1 part on the side you did pick. Getting to switch doors after 1 has been revealed is allowing you to now choose 2 doors over one when compared to the original decision. The probability is indeed 66.7% or 2/3 if you decide to switch.

The unfortunate part here is you dont get the chance to to run the switch 1,000 times in order for the percentages to even out like you would expect and more than likely get stuck on the 1/3 side with the bag of poop. LOL

I believe the percentages you are talking about are from over a course of multiple years. It’s no guarantee that each year 33% of rbs drafted whenever will become studs. Some years 0 rbs become studs

[UATahoe]

My comments had nothing to do with Rb's. LOL. Strictly on the probability theorem that was in discussion. The probabilities i reference are in direct correlation to the number of unknown variables you are dealing with. In the Monty Hall problem, that would be 3. Changing the number of variables here completely changes the principles of the theorem.

[Dynasty DeLorean]

My comments had nothing to do with Rb's. LOL. Strictly on the probability theorem that was in discussion. The probabilities i reference are in direct correlation to the number of unknown variables you are dealing with. In the Monty Hall problem, that would be 3. Changing the number of variables here completely changes the principles of the theorem.

Ok. Well I was clearly responding to the original intent of this thread and not whatever it is that you’re talking about.

[Friction]

I think Jason Werth touched on this yesterday, even if it was in reference to baseball and not football.

[UATahoe]

My comments had nothing to do with Rb's. LOL. Strictly on the probability theorem that was in discussion. The probabilities i reference are in direct correlation to the number of unknown variables you are dealing with. In the Monty Hall problem, that would be 3. Changing the number of variables here completely changes the principles of the theorem.

Ok. Well I was clearly responding to the original intent of this thread and not whatever it is that you’re talking about.

LOL noted.

[Life of Pablo]

Idk if the last few comments are talking about rookie RBs or the Monty Hall problem anymore lol. But fyi as jeffster said, the Monty Hall problem is always a legit 67% chance of winning if you understand how it works. You always want to switch doors. I'd explain it better but I'll just save time by directing to the wiki instead: https://en.wikipedia.org/wiki/Monty_Hall_problem

Having said that. I get the idea you're going for jeffster, but I don't think that problem really applies well to draft success/bust rates lol. Those are all just independent cases loosely backed by sample size. In your example of Chubb, RoJo, and another RB (let's say Sony). You pick Chubb, and are shown RoJo who busts. Monty Hall would tell you to always switch to Sony (the 67%). But even if the "success rate" of these RBs is 33%, this isn't an actual Monty Hall problem. Chubb could be a stud, Sony could be a stud, both of them could be studs, or both could bust. And whatever Chubb does, it has no effect on what Sony does (much like how RoJo busting doesn't affect either of them in the first place). The only way this would be a Monty Hall problem, is if we 100% knew for a fact that EXACTLY ONE of Chubb/RoJo/Sony would be a stud, and the other two would bust. But we don't know that at all.

TL;DR - what UATahoe said earlier.

[Mjvb5]

Key difference is in the Monty Hall case there are only three options one is correct, two are incorrect. In the case of fantasy picks one guy being a "bust" doesn't raise anyone elses hit rate as there is no exact numbers year to year, only average. For instance take the top four for 2014-2016
2014(Cooks, Watkins, Evans, OBJ?) 75%
2015(MG3, Coop, White, Gurley) 75%
2016(zeke, Coco, doctson, tread) 25%
Because of this we can't apply gurley hitting to lower gordon's rate or so but rather have to simply apply the found averages to each payer individually

[UATahoe]

We are in complete agreement and you said the same thing i did in your first paragraph.

And i agree this doesnt carry over well to rookie rb's unless, as you stated, you without a doubt are given 3 to choose from with one being 100% guaranteed to be a stud.

[Goddard]

It doesn’t really work like that. That’s like playing roulette and thinking bc it just landed on red 5 times I. A row now it’s more likely to land on black on the next roll.

I want to know when it's going to land on green.