While the following will have applications to daily games to a limited extent, I really structured this for season long.
There is a non zero probability of injury for each player. Depending on weight, height, age, biometrics and past injury history and duration of injury and recovery time vs others you can estimate a player's probability of injury and it does vary player to player. I used to have a big problem assuming "injury prone labels" but that was an objection to the old school logic of just looking at games missed and assuming the past was indicative of the future.
There's a lot more advanced data available that actually can handicap injury probability over a season.
With some help from sportsinjurypredictor.com as well as some math we can estimate the probability that at least one of a group of backup RBs suddenly see a boost in productivity due to an injury to their teammate RB.
For example, lets say we have David Johnson as RB 1 who has a low chance of injury and high chance to perform like a starting RB1 and as RB 2 we have a combination of essentially backup RBs:
Charles Sims
DeAngelo Williams
Derrick Henry
Alfred Morris
James Starks
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We know that the odds of injury of each teammate to the player over the entire season are as follows:
Sims (Martin):29%
DeAngelo Williams (Bell): 75%
Alfred Morris (Elliot): 25%
Starks(Lacy):50%
We have to use this to calculate the odds over a single game that a player will be hurt.
So we have to start with a number and some formulas and fiddle with it until we can approximate the solution.
We'll say (1-(X^16))=odds of teammates injury. X is actually going to equal the odds of no injury in a single game.
the odds of no injury times itself 16 times gives us the odds of no injury during a season. Subtracting 1 from the total gives us the chance of one or more injuries.
Subtracting 1 from X gives us the odds of an injury.
That probably isn't as easy to comprehend when put into words so I'll give you an example to simplify it.
For Sims, we find that .979^16=.712. 1-.712=.288 which is approximately a 29% chance that Martin gets injured at least once over the season.
That means you have a ~97.9% Doug Martin stays healthy week 1, or a 1-.979=~.021=2.1% chance he gets hurt.
For DeAngelo Williams Bell's odds of injury have to be condensed over 12 games instead.
.898^13=.246941 or ~25% chance Bell won't get injured over 13 games which leaves about a 75% chance that he will over the season.
This means there's about a 10.2% chance (as we find with the formula 1-.898=.102) that Bell WILL get hurt in a single game.
We can do this for all players to find the following odds player's teammates stay healthy
SIMS 0.979 DeAngelo Williams 0.898 Henry 0.9665 Morris 0.982 Starks 0.9575
Since we know we have DeAngelo Williams carrying the load for 3 weeks, we probably want to use these numbers to calculate the odds that one of these backs becomes a dominant star by week 4.
In other words, the odds that one or more of Doug Martin, Le'Veon Bell, Ezekiel Elliot or Eddie Lacy get injured one or more times by the end of week 4.
To calculate this we basically solve the odds that they all stay healthy and then subtract by 1.
For the first 3 weeks we are going to set the chances of Le'Veon Bell staying healthy at 100%. While this isn't 100% accurate, the risk of injury is primarily distributed over weeks 4-17 minus the bye week so it's easier.
so we are taking (.97*1*.9665*.982*.9575)^4 then multiplying that total times .898 (for Bell's week 4 injury risk) to get the chance everyone stays healthy all 4 weeks. If we subtract 1 minus that we get the odds that at least one will be injured.
This gives us a ~43.7% chance that we will be able to find a workhorse RB2 by the end of week 4 (not including the probability that this player is also hurt and the injury duration overlaps). Technically we need it for the week 4 game, so the odds are a bit less since this also includes those who get injured mid game or on the last snap, but if an injury happens during the week in a best ball format and that player (say James Starks) also has a role prior to (say Eddie Lacy's) injury, there's a good chance that even if the injury happens in the 3rd quarter that you'll have at least RB2 production out of the many RBs...
There's also a chance that one of those backs in a part time role can put up RB2 numbers or better that week without injury to the starter.
I think 43.7% is a fair approximation.
However... 2/3rds of injuries cause a player to be out 2 games or less. This is a problem because some of the week 1 and 2 injuries will be healthy by week 4 and some of the week 3 and 4 injuries will only last a game or two.
So we may want to take .437 times .333333333333=~.14567 or 14.57% chance that by week 4 we get a second RB that will last for more than 2 weeks.
Full Injury Breakdown!
In order to get a full injury breakdown to be more accurate and see the probability that we have an RB2 who is a workhorse back due to injury to the RB1, we have to compile all of the stats on how long injuries last.
What's the probability that an injury puts a player out exactly 1 week? exactly 2? exactly 3?
If we set conditions and formulas and run a monte carlo simulation we can simulate the season with these RBs and consider the probability that an individual possible outcome or group of outcomes occurs.
While simulating or calculating whether or not an injury occurs or not is useful, we ultimately need to determine how long this lasts, and have an approximate idea of our point total when it does and when it doesn't to be able to approximate season point totals and determine our probability of reaching a threshold we believe is enough to win... (alternatively simulating every single player's health and expected points is another more complicated option.)
So far we've only considered the odds that a player in front of an RB on the depth chart opens up an opportunity for some backup, but we did not yet include the chance that the backup on our team is also hurt during this time period.
The point of all of this is that a team where you only draft the minimum number of players you need at each position will be weaker at depth and at risk to injury. The team that intentionally neglects at least one position initially but makes up for it in depth will likely be the best IF a few things go their way such as the injuries lining up OR a player earning a larger number of carries than anticipated for other reasons.
You can instead try to calculate the odds that everyone on your team stays healthy by a similar method
Ultimately you want the strategy that has the greater probability of acheiving your goal. To a winner take all format that adds up all points at the end of the year with no playoffs, the goal is winning and so getting last place 85% of the time and 1st place 15% of the time would actually be very successful as the average team in a 12 team league wins 1/12=~8.33%.
All of this is a tremendous amount of work, but for those willing, you can at least see the logic of it.
GPP tournament players in daily games? The next post is for you...
Since players are very unlikely to use a 2nd or 3rd RB, playing an unusual player like this is very contrarian. Even though it's a low probability bet, betting on a #2 or even #3 RB will be very contrarian and if he gets injured EARLY AND also is the most productive back in the league that week, you will be ahead by a large margin at that position, and in terms of salary, the cost will be incredibly low allowing you to bet on other higher probability plays. We will explore this tactic next.
