How to calculate risk of death over career life?

Let's say for a certain occupation, there are 1M in that job and 100 of them die every year on the job, because of the job.

The one-year risk of death is low (100 ÷ 1M = 0.01%, or 1 in 10k).

But what's the risk over 30 years? I can think of two ways to approach it.

First, if 100 die in 1 year, then over 30 years, 3000 will die. 3000 ÷ 1M = 0.3%, or 1 in 333.

Yes, it's not the same exact 1M workers over that 30 years (some will retire, or die, and others enter the workforce), but the total number will be pretty stable, so it seems like that method should work.

The other way is to look at the chances of an event happening with repeated trials:

1-(chances of event not happening ÷ total chances)^trials

1 - ( 9999 ÷ 10,000 )³⁰ = 0.003, or 1 in 333.

This is an unexpected result for me. One in 10k seems really long, and I'm surprised it got whittled down to a mere 1 in 333 (which seems really short) after repeating for 30Y. But maybe I erred in my calculations, or I'm missing something else. Am I missing something?

Quote: MichaelBluejay

Let's say for a certain occupation, there are 1M in that job and 100 of them die every year on the job, because of the job.

The one-year risk of death is low (100 ÷ 1M = 0.01%, or 1 in 10k).

But what's the risk over 30 years? I can think of two ways to approach it.

First, if 100 die in 1 year, then over 30 years, 3000 will die. 3000 ÷ 1M = 0.3%, or 1 in 333.

Yes, it's not the same exact 1M workers over that 30 years (some will retire, or die, and others enter the workforce), but the total number will be pretty stable, so it seems like that method should work.

The other way is to look at the chances of an event happening with repeated trials:

1-(chances of event not happening ÷ total chances)^trials

1 - ( 9999 ÷ 10,000 )³⁰ = 0.003, or 1 in 333.

This is an unexpected result for me. One in 10k seems really long, and I'm surprised it got whittled down to a mere 1 in 333 (which seems really short) after repeating for 30Y. But maybe I erred in my calculations, or I'm missing something else. Am I missing something?
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It appears, that it is intuition only, which again is misleading human thinking when trying to estimate probabilistic relationships.
By pure coincidence, I stumbled over this video, yesterday:


Starting at about 43:25 they discuss the probability of getting breast cancer and it is interesting to see how drastically the probability rises over longer periods of life time.

PS: Back in December 2015, discflicker already pointed out to this video..

Quote: MichaelBluejay

Let's say for a certain occupation, there are 1M in that job and 100 of them die every year on the job, because of the job.

The one-year risk of death is low (100 ÷ 1M = 0.01%, or 1 in 10k).

But what's the risk over 30 years? I can think of two ways to approach it.

First, if 100 die in 1 year, then over 30 years, 3000 will die. 3000 ÷ 1M = 0.3%, or 1 in 333.

Yes, it's not the same exact 1M workers over that 30 years (some will retire, or die, and others enter the workforce), but the total number will be pretty stable, so it seems like that method should work.

The other way is to look at the chances of an event happening with repeated trials:

1-(chances of event not happening ÷ total chances)^trials

1 - ( 9999 ÷ 10,000 )³⁰ = 0.003, or 1 in 333.

This is an unexpected result for me. One in 10k seems really long, and I'm surprised it got whittled down to a mere 1 in 333 (which seems really short) after repeating for 30Y. But maybe I erred in my calculations, or I'm missing something else. Am I missing something?
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Let's see...

I can think of a third way; let's look at the probability of the event NOT happening for one year and then extrapolate that to thirty years. (This is basically the same as your second way; it's just expressed slightly different.)

999900/1000000 = 0.9999

0.999^30 = 0.99700434594

Inverse: 1 - 0.99700434594 = 0.00299565405

1/0.00299565405 = 1 in 333.81691721

It would seem that our results agree. I don't know if this is a thought experiment or something that you're actually doing on a professional sort of basis, but I would think the odds for one specific individual to die on the job, over thirty years, could generally be construed to be shorter than this. Obviously, this would vary based on the occupation, but given the fact that one can generally be expected to be in worse overall health in year 30, as compared to year 1, I would say that a specific individual in his 30th year would be more likely to die than he was in his first year. In the meantime, some of the people who did, in fact, die, didn't make it to their 30th year on the job in the first place.

Of course, that would be difficult to account for and more information would be needed. Specifically, we would want to look at death statistics for that particular occupation and verify that there is a positive correlation between years on job and probability of death. Depending on the job, there's also a chance that a lack of experience in that job could be a contributing factor to some deaths, so we might find that the probability of death correlates positively with both having less experience AND being more advanced in age...which is to say it skews to the ends of the 1-30 range. Naturally, you'd need the specific death statistics to make such a determination---and even then, it would arguably be too limited a sample size.

I don't think I explained my thought process as well as I might have, so allow me to clarify:

When we talk about 100 per 1,000,000, per annum, we're discussing the entire population of those who work this position. My suggestion is that this information is insufficient to get to a thirty-year probability of death by way of a workplace incident as we have insufficient information.

Doing it this way would only be sufficient if we were only looking at people (dead ones aside) who worked the full thirty years. In general terms, I would want to pull samples from the population to determine if death is more likely, in a given year, for those samples. Specifically, I might look at:

Years 0(i.e. Day One)-5
Years 6-10
Years 11-15
Years 16-20
Years 21-25
Years 26-30

My working hypothesis would be (depending on position) that some deaths in the early years might be due to inexperience whereas death in later years might be due to health at the time of the incident; while experience might lead to a lesser probability to be impacted by such an incident, older people might be less likely to survive an incident if there is one.

If we create an example wherein the year 0-5 probability is, say, 2% and the years 26-30 probability is 1.5%, then what we would find is that years 6-25 are sufficiently low, on balance, to get us to our 1% overall. However, the next relevant consideration would be average length of time working this position.

If we find that the average length of time working this position is twenty years, then we would have to treat a thirty year career differently. While someone working a twenty year career has gone through (obviously) the first five years, they spent the last fifteen years ducking the other high-correlation time for the purposes of our example, which is Years 26-30.

With that, what we would have to do is go through and compare the sample (years on job) range to the percentage who die for each of the six steps; at that point, we would arrive at something closer to the probability of death for someone to actually work thirty years at this position.

On the other hand, it might turn out that death (relative to time on job) is, actually, distributed with relative uniformity. I'd have a harder or easier time believing that depending on what the job is.

Thank you, this is helpful.

I was looking into this because someone I know is contemplating entering a dangerous profession and asked me about the odds.

On a related note, here are some 1 in X odds of dying in one year from the following in the U.S., using the latest data (2022 or 2023, whichever was available):

1 in 7800 • Motor vehicle (n=43,000)
1 in 18,000 • Firearms (excluding suicide) (n=19,000)

U.S. population is 336M. So lifetime (77Y) risk for dying for each is:

1 in 102 • Motor vehicle
1 in 230 • Firearms

This odds are so short that by middle age, just about everyone will know someone who died from each. I do. Anyone here who *doesn't* know someone who died from each?

These numbers are simple to calculate, but even people with post-graduate degrees get it wrong. For example:

"According to the National Safety Council (NSC), your chances of dying in a car crash are about one in 103 for any given year." (Rosenfeld Injury Lawyers) No, that's LIFETIME risk, not ONE YEAR risk.

Others (e.g. Rice Law MD) refer to the risk ("1 in 107") without bothering to specify what time period it's for.

I found this because Googling "chances of dying in a car crash in x years", the first dozens of results are from law firms writing throwaway articles for clickbait.

No wonder they have to pay for expert testimonies!

I’ve personally known three people to die by way of motor vehicle (same incident for two of them), but haven’t known anyone involved in a firearms-related death, if we’re excluding suicide.

I wouldn't think the odds would be that high after 30 years.... After that much time you'll have a lot more experience on the job, and you'll know what mistakes to avoid and how to avoid them. Someone with, say, two years of experience with heavy machinery is probably a lot more likely to die than someone who's been around that equipment for 30 years and can operate it in their sleep. Also, I would hope that someone who is in a profession for 30 has moved up the ranks somewhat into some kind of management or office position, and is not around some of the more dangerous elements.

Those 100 people dying on the job every year out of a million are probably not the old-timers and guys close to retirement. It's the new and inexperienced guys, and some guys who just get involved in freak accidents. If anything, I bet the odds of dying are even less than 1 in 10k for those 20-30 year veterans. Just look at the military; most of the deaths (during wartime, anyway) are combat deaths or accidents involving younger guys. Once you've been in the military for 20-30 years, unless you actively seek out a dangerous position, you're pushing pencils behind a desk well behind the front lines or in another country altogether. Make it 40 years in the military, and you're probably at the Pentagon or White House or some other super safe location.

Quote: Mission146

No wonder they have to pay for expert testimonies!

I’ve personally known three people to die by way of motor vehicle (same incident for two of them),

How many of those three were you responsible for mowing down? Ex-wives will not be held against you.

Quote: DRich

Quote: Mission146

No wonder they have to pay for expert testimonies!

I’ve personally known three people to die by way of motor vehicle (same incident for two of them),

How many of those three were you responsible for mowing down? Ex-wives will not be held against you.
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The two that were the same incident; I'm nothing if not efficient.

The plot thickens. So, the occupation in question is firefighter (FF). I've run into a paradox: The chances of line-of-duty death (LODD) seems to be high, but studies say that FF have *lower* mortality than the general population.

Earlier I'd said 100 out of 1M LODD/Y. Let's use more precise numbers: The average # of U.S. LODD over the last 10Y is 70 (Statista), and I'll presume that the number of U.S. FF is pretty stable at 1,057,600 (FEMA).

That puts the 1-year odds at 1 in 15k, career (30Y) odds at 1 in 504.
Given that the 30Y risk of dying in a car crash is 1 in 252 (as 44k/Y of 332M), the *additional* 1 in 504 risk looks to be pretty significant to me. You don't have chance of dying of crash/cancer/heart disease/etc. *OR* FF, by being a FF, you've added an *additional* risk of death, because as a FF you could still die from crash/cancer/heart disease/etc.

Okay, 15% of LODD are crashes (NFPA), and we can't double-dip. So excluding LODD crashes, LODD risk is 1 in 561. Still seemingly significant.

The only two studies I could find that looked at all-cause mortality for FF showed that their mortality is 1% and 10% less than for the general population. (Johns Hopkins 2005, Occ. & Env. Med. 2014). (The first is a meta analysis; I'm in the process of tracking down the individual studies they relied on, and checking the methodology of all the studies, but I presume that they normalized the age of the referent population.)

If FF have an *added* risk burden, this doesn't seem possible, unless being a FF makes them *way* less likely to die from other typical causes of death. In fact, at least one study suggested that possibility, calling it the "Healthy Worker Effect" (HWE), that FF might be healthier than the general population and thus less likely to die of other causes. But that answer seems really unsatisfying. According to Rip Esselstyn, who had a NY Times bestseller (The Engine 2 Diet) about getting his station crew to switch to healthy eating, the typical firehouse diet is exceptionally poor.

So, is there truly a paradox, or am I missing something, or does HWE really explain the apparent paradox?

Quote: MichaelBluejay

The other way is to look at the chances of an event happening with repeated trials:

1-(chances of event not happening ÷ total chances)^trials

1 - ( 9999 ÷ 10,000 )³⁰ = 0.003, or 1 in 333.
link to original post

That's how I would look at it. Except I round to 1 in 334.

Quote: Mission146

I can think of a third way; let's look at the probability of the event NOT happening for one year and then extrapolate that to thirty years. (This is basically the same as your second way; it's just expressed slightly different.)

I think you are on the right track but I would suggest it a slightly different way.

I have to leave for work shortly, so this is going to be brief.

IMHO the thing to do it take the math and see how long it would take to happen. My example is an article I read once of a woman who wanted to defend "free ranging" her kids. People she knew bombarded her with the "all these kids being snatched" thing. So she turned it back by taking the number of kids taken not by family members, which really cut it back. (Bear with me here) Then she figured how long in years it would take for I forget the exact but say a 50% chance the kids would be snatched. It turned out to be something like 180 years.

So to the firefighter thing, figure how long for a 50% chance of a fatality, no matter how long it takes. Then you need to see how the particular job compares. A forestry firefighter might well have a higher fatality rate. You may want to back out the 2001 outlier year where hundreds died in one event. Do all that and get to how many years you need to work the job for a statistical 50% chance.

Quote: MichaelBluejay

The plot thickens. So, the occupation in question is firefighter (FF). I've run into a paradox: The chances of line-of-duty death (LODD) seems to be high, but studies say that FF have *lower* mortality than the general population.

Earlier I'd said 100 out of 1M LODD/Y. Let's use more precise numbers: The average # of U.S. LODD over the last 10Y is 70 (Statista), and I'll presume that the number of U.S. FF is pretty stable at 1,057,600 (FEMA).

That puts the 1-year odds at 1 in 15k, career (30Y) odds at 1 in 504.
Given that the 30Y risk of dying in a car crash is 1 in 252 (as 44k/Y of 332M), the *additional* 1 in 504 risk looks to be pretty significant to me. You don't have chance of dying of crash/cancer/heart disease/etc. *OR* FF, by being a FF, you've added an *additional* risk of death, because as a FF you could still die from crash/cancer/heart disease/etc.

Okay, 15% of LODD are crashes (NFPA), and we can't double-dip. So excluding LODD crashes, LODD risk is 1 in 561. Still seemingly significant.

The only two studies I could find that looked at all-cause mortality for FF showed that their mortality is 1% and 10% less than for the general population. (Johns Hopkins 2005, Occ. & Env. Med. 2014). (The first is a meta analysis; I'm in the process of tracking down the individual studies they relied on, and checking the methodology of all the studies, but I presume that they normalized the age of the referent population.)

If FF have an *added* risk burden, this doesn't seem possible, unless being a FF makes them *way* less likely to die from other typical causes of death. In fact, at least one study suggested that possibility, calling it the "Healthy Worker Effect" (HWE), that FF might be healthier than the general population and thus less likely to die of other causes. But that answer seems really unsatisfying. According to Rip Esselstyn, who had a NY Times bestseller (The Engine 2 Diet) about getting his station crew to switch to healthy eating, the typical firehouse diet is exceptionally poor.

So, is there truly a paradox, or am I missing something, or does HWE really explain the apparent paradox?
link to original post

I think firefighters, by the nature of the job being so physical, and I’m pretty sure there are weight requirements (no obese need apply?) , would tend to be healthier than the average Joe or Jane.
Friends son just became a firefighter right out of high school. Also trained to be an EMT. Making GREAT $$ for a 19 year old kid. But I’m sure momma is worried…..

Quote: SOOPOO

I think firefighters, by the nature of the job being so physical, and I’m pretty sure there are weight requirements (no obese need apply?) , would tend to be healthier than the average Joe or Jane.
Friends son just became a firefighter right out of high school. Also trained to be an EMT. Making GREAT $$ for a 19 year old kid. But I’m sure momma is worried…..
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I always assumed firefighters were required to have EMT training. About 30 years ago I had a firefighting textbook. I assume it is a little out of date now.

I think you may be massively undervaluing the healthy worker effect for people who routinely carry 60 pounds of extra weight and run up 5 flights of stairs.

Quote: SOOPOO

Quote: MichaelBluejay

The plot thickens. So, the occupation in question is firefighter (FF). I've run into a paradox: The chances of line-of-duty death (LODD) seems to be high, but studies say that FF have *lower* mortality than the general population.

Earlier I'd said 100 out of 1M LODD/Y. Let's use more precise numbers: The average # of U.S. LODD over the last 10Y is 70 (Statista), and I'll presume that the number of U.S. FF is pretty stable at 1,057,600 (FEMA).

That puts the 1-year odds at 1 in 15k, career (30Y) odds at 1 in 504.
Given that the 30Y risk of dying in a car crash is 1 in 252 (as 44k/Y of 332M), the *additional* 1 in 504 risk looks to be pretty significant to me. You don't have chance of dying of crash/cancer/heart disease/etc. *OR* FF, by being a FF, you've added an *additional* risk of death, because as a FF you could still die from crash/cancer/heart disease/etc.

Okay, 15% of LODD are crashes (NFPA), and we can't double-dip. So excluding LODD crashes, LODD risk is 1 in 561. Still seemingly significant.

The only two studies I could find that looked at all-cause mortality for FF showed that their mortality is 1% and 10% less than for the general population. (Johns Hopkins 2005, Occ. & Env. Med. 2014). (The first is a meta analysis; I'm in the process of tracking down the individual studies they relied on, and checking the methodology of all the studies, but I presume that they normalized the age of the referent population.)

If FF have an *added* risk burden, this doesn't seem possible, unless being a FF makes them *way* less likely to die from other typical causes of death. In fact, at least one study suggested that possibility, calling it the "Healthy Worker Effect" (HWE), that FF might be healthier than the general population and thus less likely to die of other causes. But that answer seems really unsatisfying. According to Rip Esselstyn, who had a NY Times bestseller (The Engine 2 Diet) about getting his station crew to switch to healthy eating, the typical firehouse diet is exceptionally poor.

So, is there truly a paradox, or am I missing something, or does HWE really explain the apparent paradox?
link to original post

I think firefighters, by the nature of the job being so physical, and I’m pretty sure there are weight requirements (no obese need apply?) , would tend to be healthier than the average Joe or Jane.
Friends son just became a firefighter right out of high school. Also trained to be an EMT. Making GREAT $$ for a 19 year old kid. But I’m sure momma is worried…..
link to original post

Obese might matter different than you think. These are usually big dudes carrying lots of equipment. I don’t think the look as much like some WWE superstar as they do the big guy at the bar. Do out of shape obese no. Hockey goon maybe.