expected lost games to injury

Can any one tell me if my math is OK

I have a player with a high injury rating of 5, how may games in an average season will he miss--I know a 5 will be hit about 15 times a year on a no injury card (for a full year of 648 rolls in 162 games) I figure on an injury roll of 5 will be hit about once every 11 played games or so--average games missed per injury I guess around 4 games--so if 4 missed games every 15 games then about 118 games played? (162 / 15 * 11) =118.8. This seems a bit optimistic to me.

5 years later edit--that should have read ((162/15) * 11)

I don't know how accurate my method is but I figure 6% time lost for each injury chance. so with a 5 youre looking at 30% games missed
hope that helps also hope someone better at this than me chimes in!
SF

30% loss to 162 = 113 games, so I guess I am "in the ballpark" with 118 and your method is far easier to figure.

Just did the math from the chart, counting the "rest of game " as a half game, your average injury would be 3.55 games per 216 plate appearances per injury chance. Assuming the typical batter come to bat 4.5 times per game( this is just a ballpark guess) that would be 729 PA per season which would be 12 injury days per chance, or 7.4 percent
I am fairly comfortable with this conclusion

"starts missed" is different from "games missed" -- guys get injured while in the game, which is a partial game missed

defining "game" as "9 innings" (so being injured for rest of game at the mid=point would equal "one game" if it happened twice), the short-hand rule for fewer than 600 AB+W is this:

(23-inj)x(inj/2)

a 5 inj is thus 18x2.5= 45 (i.e., 45 x 9 innings missed), with a small margin of error the higher the rating

of course, the higher the rating, the greater the standard deviation -- the above formula is the average, without regard to standard deviation

it varies a lot if the guy plays every day and leads off, compared to a platoon guy who bats ninth

here's the math (from an old post):

FORMULA FOR AVERAGE NUMBER OF STARTS MISSED, DEPENDING ON INJURY RATING:

(BIA/C) / (1 + (BIA/EC) + 0.9(DB/EC))

"/" means "divided by"; unseparated letters are multiplied

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FORMULA FOR AVERAGE NUMBER OF GAMES MISSED, DEPENDING ON INJURY RATING:

This accounts for the "remainder of game" amount that's tacked on to every injury.

It's easier to think of "games missed" this way:

GAMES MISSED = "STARTS MISSED" TIMES (1 + (D/A))

The full formula (which just inserts the formula instead of the words "starts missed") is:

((BIA/C) / (1 + (BIA/EC) + 0.9(DB/EC)))(1 + (D/A))

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For the meaning of the letters in the formula, scroll down. Except for "A" and "I," the values can be modified based on your preferences. However, "E" (games in season) must be 162 for the formula to work perfectly; other numbers will introduce a slight error. (Note: Personally, I think "C" should always be 216.)

One remaining problem, or maybe not a problem: In any 162 game season, the actual limit of starts missed due to injury (though it would never happen) is 151, not 162. This fact (i.e., that a range of zero to 151 isn't the same as zero to 162) probably means that all results of the formula should be slightly lowered, perhaps by three percent, give or take. I don't know -- it's also possible that the effect is already baked into the formula.

A AVERAGE NUMBER OF STARTS MISSED PER INJURY
B AVERAGE NUMBER OF FULL-SEASON PAs ("EVERY-INNING" PLAYER)
C 216 (OR PREFERRED SLIGHTLY HIGHER NUMBER)
D "REMAINDER OF GAME" AVERAGE VALUE (MAXIMUM 0.5)
E GAMES IN SEASON
I INJURY RATING

A = 3.45432062 FOR SUB-600 AB+W PLAYERS
A = 1.82932095 FOR 600-PLUS AB+W PLAYERS

(These weird numbers account for the fact that end-of-season injuries are shortened if the season ends before the injury "runs out," thus lowering the overall average. I give the full numbers for spread-sheets; obviously, you can remove the last five decimals otherwise.)

B = your estimate of how many PAs an every-inning full-season player "would" have, on average (if preferred, note that full-time lead-off hitters have more than average, full-time 9th hitters fewer, etc.; also, slugging teams have more PAs on average, while low-scoring teams have fewer; and non-pitching starters will average slightly more in no-DH leagues). This number is for a mythical "ideal" player who is in every single moment of every game. This number excludes sacrifice bunts and H&Rs.

C = 216 if you think the long-term probability of any roll is a multiple of 1/216; use a slightly higher number if you think injuries are proportionally less likely than other types of "plays"

D = on average, how many "games" does "remainder of game" equal? The maximum possible is 0.5; the actual number is 0.47 or lower, depending on how you define the value of "remainder of game" immediately after a plate appearance has occurred. It would be irrational to use a number lower than, say, 0.25.

E = 162 for an opening-day-roster regular season; 174 for the maximum through game 7 of the finals. The number can also reflect a mid-season replacement. NOTE: any number different from 162 will introduce a slight error in "average starts missed per injury," the "A" in the formula: "A" will slightly increase for numbers between 163-174, somwewhat decrease for numbers lower than 162, and dramatically decrease for numbers substantially lower than 162. (That's due to end-of-season injuries factoring into the average, because sometimes the season ends before the injury "runs out"; this lowers the overall regular-season average in general, but especially for mid-season replacements.)

I = injury rating, from 1 to 6.
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Here's an example, using 670 for "B" (average full-season PAs) and 0.42 for "D" (average value of "remainder of game"). I left "C" as 216 and "E" (length of season) as 162. ("A" and "I" are not variable.) You can use other numbers if you prefer, with no change in the formula's strength except if you vary "E" substantially.

(If any of the below varies from what the formula "should" produce, then it just means I've made a number-crunching error -- I didn't double-check.)

Average starts and games missed per season for "sub-600 AB+W" players:

rating 1 -- 9.98 starts missed and 11.20 games missed
2 -- 18.81 and 21.09
3 -- 26.66 and 29.90
4 -- 33.70 and 37.80
5 -- 40.04 and 44.91
6 -- 45.79 and 51.35

Average starts and games missed per season for "600 or more AB+W" players:

1 -- 5.44 and 6.69
2 -- 10.53 and 12.95

amazing work Clowntime! I have one question, why would a player with 2 injury chances not simply miss twice as many games as a player with 1 injury chance?

at the very bottom of the post you can see that the ratio is nearly 2-1

the reason it's less is: injured players sit out games, and so have fewer chances to get injured

the formula basically imagines an "uninjurable" player, and then DISCOUNTS that player's total PAs based on injury rating

a 2 injury rating will have fewer chances to get injured than a 1 player, so on the margins the "discount" is very slightly less

also, the end-of-year factor gets increasingly important as the inj rating gets higher -- if a guy is injured in game 162, the most he can be injured for is rest of game (the formula allows an adjustment if you want to include post-season, so say game 174 if you want) -- this also slightly favors higher injury ratings, so the ratio does not go 6-5-4-3-2-1, but slightly lower

note the formula can be greatly simplified if you accept 1 in 216 as the actual odds (some people don't), and define "rest of game" as an average of, say, 0.47 games, which is undoubtedly very close; and presume 162 games, and just substitute the constant numbers (e.g., "A" -- average games missed per injury, which is undebatable unless the game engine has a secret rule)

wow thanks. When I referred to you as "the injury guy" in the league we're in , little did I know how true that was!

well, my new team is partly a small sample test of a thing I believe -- HAL "penalizes" expensive players more than cheap players for the same injury rating -- Longoria starting every game will get injured more than a 1.0m guy starting every game with the same injury rating -- but who knows

I am playing my first draft season --making many rookie mistakes -- so thanks Steve & Clowntime--and I though my method was fuzzy---but again 162-44.91(your average games missed for a 5 injury) =117.09 played, which is real close to both of our guestimates of 113 and 118 played--so in essence a 5 injury player is in just over 2/3rds of a season--so if you have a decent replacement he may be worth the risk. I'm really surprised they play that much. (player I'm thinking about is 2012 3rd baseman Longoria )

This doesn't include carrying two injury prone catchers as I have read in other posts the game engine will not let both catchers get hurt simultaneously, sometimes making two injury prone catchers who both hit well an advantage, as if one is injured the other cannot be until the first is off injury list.