What Do You Make Of This...Injuries

Here is another example of case in point:
KEEP IN MIND HES a 1 on injury chances- yet Dimaggio has gotten hurt 4 times in 70 games- how can Elston, Cronin 2X's and Belle be the ONLY other players that got hurt?
Now this is not statistically impossible- but its very unlikely- yet it seemingly happens very regularly...

Player Pos Risk Salary Injured during For
TOTAL 35 days
Howard, Elston (1962) C 2 3.48M Game 1 7 additional days
Dimaggio, Joe (1939) CF 1 10.01M Game 4 4 additional days
Dimaggio, Joe (1939) CF 1 10.01M Game 10 3 additional days
Cronin, Joe (1941) SS 1 5.42M Game 22 3 additional days
Dimaggio, Joe (1939) CF 1 10.01M Game 25 10 additional days
Cronin, Joe (1941) SS 1 5.42M Game 48 3 additional days
Dimaggio, Joe (1939) CF 1 10.01M Game 57 2 additional days
Belle, Albert (1995) LF 1 7.00M Game 63 3 additional days

If we use the simplifying assumptions that rburgh introduced above then we can calculate how likely these outcomes are. Its straightforward to calculate this using the BINOMDIST function in Excel. Assuming 3 injury rolls each season for a 2/12 chance guy gives a per game injury probability of 0.0185 for each player. Joe Dimaggio's chance of getting injured 4 times in 70 games is about 4%. However, the chance of any one player getting injured 4 times in 70 games is much higher - over 30%. So this is not a particularly unlikely outcome.

You remark that only 4 positions have been injured in 70 games. This is a relatively rare occurrence. Using the same assumption, it should only happen about 5% of the time. However this can mostly be attributed to the fact that you have only had 8 injuries in 70 games whereas, by rburgh's assumptions, the most likely number is 11 or 12. The probability of having only 8 injuries (or less) is 15%.

To compare apples to apples, I'll post my current teams through 70 games. Team 1:
Lajoie, Nap (1901) 2B 1 10.64M Game 1 5 additional days
Walker, Larry (1999) RF 2 9.48M Game 15 1 additional day
Walker, Larry (1999) RF 2 9.48M Game 18 2 additional days
Pujols, Albert (2008) 1B 1 10.51M Game 17 3 additional days
Duffy, Hugh (1901) CF 2 .90M Game 15 5 additional days
Pujols, Albert (2008) 1B 1 10.51M Game 43 1 additional day
Wells, Willie (NeL) SS 1 10.03M Game 43 3 additional days
Pujols, Albert (2008) 1B 1 10.51M Game 49 1 additional day
Ward, John (1889) SS 1 .85M Game 46 4 additional days
Beckwith, John (NeL) 3B 1 9.90M Game 50 3 additional days
Wells, Willie (NeL) SS 1 10.03M Game 54 1 additional day
Santo, Ron (1964) 3B 1 7.80M Game 62 2 additional days
Lajoie, Nap (1901) 2B 1 10.64M Game 70 2 additional days
- Nothing odd here, no one has been injured more than 3 times and 7 of the 9 positions have been injured (Beckwith is my DH). You might remark on Pujols getting injured 3 times. The odds of any one of nine 2/12 chance guys getting injured 3 times in 70 games is almost 75%.

Team 2:
Carter, Gary (1982) C 1 7.69M Game 27 3 additional days
Larkin, Barry (1996) SS 1 10.88M Game 31 3 additional days
Bonds, Barry (2001) LF 1 14.54M Game 34 1 additional day
Carter, Gary (1982) C 1 7.69M Game 41 1 additional day
- Only 4 injuries in 70 games. Now this would seem to be very rare, but its because I'm playing Roger Conner, Craig Biggio, and David Wright, who can't get injured. The odds are about 10% for this to occur.

Team 3:
Brown, Willard (NeL) CF 1 8.55M Game 2 1 additional day
Dimaggio, Joe (1941) CF 1 11.57M Game 4 1 additional day
Brosius, Scott (1998) 3B 1 5.59M Game 2 4 additional days
Wells, Willie (NeL) SS 1 10.03M Game 4 2 additional days
Santo, Ron (1964) 3B 1 7.80M Game 4 2 additional days
Ortiz, David (2007) 1B 1 7.35M Game 17 2 additional days
Henderson, Rickey (1990) LF 1 8.75M Game 33 1 additional day
Walker, Larry (1997) RF 1 10.85M Game 43 3 additional days
Dimaggio, Joe (1941) CF 1 11.57M Game 48 2 additional days
Ortiz, David (2007) 1B 1 7.35M Game 49 3 additional days
Henderson, Rickey (1990) LF 1 8.75M Game 57 3 additional days
Lajoie, Nap (1901) 2B 1 10.64M Game 60 1 additional day
Walker, Larry (1997) RF 1 10.85M Game 67 1 additional day
- Pretty likely outcome. No one has been injured more than twice and 7 of 8 positions have been injured (Frank Thomas is my first baseman).

You can draw what conclusions you like. Bear in mind that focusing on individual outcomes within a random distribution can be misleading.

Thanks for the help guys.

djmacb, if I'm understanding you correctly you are saying one of nine players (in this case Pujols) getting hurt 3 times in 70 games which at that rate means 7 times for a season...is about 75% likely? ..seems high....and based on 1-3 game injury potential, on average he'd miss about 14 games (somewhere between 7-21 games) so Pujols on 2/12 getting hurt 7 times in about 148 games should be unlikely.

I think what Charlie and others are saying is if every 2/12 injury should get hurt 3 times a season....2 or 3 might not get hurt at all, and two or three might get hurt 5,6,7 times...and it averages out. I understand..I agree....BUT it should happen like that "sometimes"...and it seems to happen like that "most" times.

Thanks for the help guys.

djmacb, if I'm understanding you correctly you are saying one of nine players (in this case Pujols) getting hurt 3 times in 70 games which at that rate means 7 times for a season...is about 75% likely? ..seems high....and based on 1-3 game injury potential, on average he'd miss about 14 games (somewhere between 7-21 games) so Pujols on 2/12 getting hurt 7 times in about 148 games should be unlikely.

Nev, I'm saying that for one particular player to get hurt 3 times (or more) in 70 games the probability is 15%,but for any of the 9 players to get hurt 3 times or more the probability is 75%. Its a matter of combinatorics. Look at it this way - in order for no players to get hurt 3 times in 70 games, that means each and every player must be injured 2 times or less. That is less probable than one of the 9 getting hurt 3 times.

I think what Charlie and others are saying is if every 2/12 injury should get hurt 3 times a season....2 or 3 might not get hurt at all, and two or three might get hurt 5,6,7 times...and it averages out. I understand..I agree....BUT it should happen like that "sometimes"...and it seems to happen like that "most" times.

It is much, much, MUCH more probable that two or three guys get hurt 5, 6, or 7 times, than for all of your players to be hurt the same number of times (3). Again, it's a matter of combinatorics. I'll try to explain it this way - its like dropping identical balls (27 injuries) into 9 baskets (positions - C,1B,2B,SS,3B,LF,RF,CF,DH). How many ways can you arrange the 27 balls so that each basket gets 3 balls - the answer is only one way. How many ways can you arrange the 27 balls so that one position gets all 27 - the answer is 9 ways, each basket having all 27 in its turn. Therefore - this is counterintuitive - but it's 9 times more probably that one position gets all 27 injuries than that the 27 are shared exactly equally between all 9 positions.

Similarly, its 81 times more probably to have an injury distribution of 4,2,3,3,3,3,3,3,3 for the 9 positions than 3,3,3,3,3,3,3,3,3. Again this is due to combinatorics. For other distributions, the math is beyond me (Hal Richman can probably weigh in, ha ha), but the critical point is that an exactly equal distribution of injuries is very, very unlikely.

To restate what dj said:

Average player gets about 700 PAs in a season. 1/2 are on hitters' card, so 350 ABs. If he has a 1/108 chance on every hitter roll of getting hurt, that falls between 3-4 injuries per season.

Assuming 4.5 PAs per game, 2.25 on hitter card. 108 chances / 2.25 PAs per game means an injury on average every 48 games.

That doesn't address the player in multiple leagues argument, but it Stratifies the math a bit.

I think what Charlie and others are saying is if every 2/12 injury should get hurt 3 times a season....2 or 3 might not get hurt at all, and two or three might get hurt 5,6,7 times...and it averages out. I understand..I agree....BUT it should happen like that "sometimes"...and it seems to happen like that "most" times.

It is much, much, MUCH more probable that two or three guys get hurt 5, 6, or 7 times, than for all of your players to be hurt the same number of times (3). Again, it's a matter of combinatorics. I'll try to explain it this way - its like dropping identical balls (27 injuries) into 9 baskets (positions - C,1B,2B,SS,3B,LF,RF,CF,DH). How many ways can you arrange the 27 balls so that each basket gets 3 balls - the answer is only one way. How many ways can you arrange the 27 balls so that one position gets all 27 - the answer is 9 ways, each basket having all 27 in its turn. Therefore - this is counterintuitive - but it's 9 times more probably that one position gets all 27 injuries than that the 27 are shared exactly equally between all 9 positions.

Similarly, its 81 times more probably to have an injury distribution of 4,2,3,3,3,3,3,3,3 for the 9 positions than 3,3,3,3,3,3,3,3,3. Again this is due to combinatorics. For other distributions, the math is beyond me (Hal Richman can probably weigh in, ha ha), but the critical point is that an exactly equal distribution of injuries is very, very unlikely.

and herein lies the problem-
No one is expecting an exact distribution, or really a distribution where everyone is within one injury or two even.

Its the recurrence of injuries to one single player combined with the exclusion of MANY of the other players entirely.

If it were anything like 4,2,3,3,1,1 etc. as you say, that makes total sense. This does not.

I cant stress enough that the 3% likely scenario appears to be happening a preponderance of the time.

EVEN Further is the injury shown to the pitcher- sure the DH can be injured 4 times in one season- but what are the odds that it happens when hes At Bat with the VERY SAME pitcher each time- and no times with any other starter or reliever

I'm certainly not the sharpest mind in this conversation, but The Last Druid and Salty are both pretty sharp and these examples haven't changed their mind...and that leaves some doubt in mine too...Regardless I really appreciate such attempts to dumb it down for me.

I am not a math whiz by any means, but I agree 100% with dj. He explained it far better then I could ever manage.

Funny part is I have long suspected that you are more susceptible to injury directly after coming off one.

EVEN Further is the injury shown to the pitcher- sure the DH can be injured 4 times in one season- but what are the odds that it happens when hes At Bat with the VERY SAME pitcher each time- and no times with any other starter or reliever

Can you explain what you mean or show some examples? Its not clear to me what you are trying to say. The DH is injured off of his card. The pitcher is injured on a 6-12 roll (pitcher's card) when the other team's DH is at bat.