Probability given an occurrence rate (small airplane fatal accidents)

C

Cloud8

New member
Joined
Nov 11, 2019
Messages
6
Hi All,

I would like to know if the method I’m using to calculate a probability based on an occurrence rate is correct or if there is a better method to use.

Here is some context to give you an idea of what I’m working with: I am looking at determining the probability of a fatal accident occurring in a small airplane within the next 20 years vs number of hours flown. I am using historical data from 1974-2016 (2016 is the latest year available) to project rates into the future using an exponential trend line. I am then adjusting these numbers for a variety of reasons (but that is beside the point).

Here are the projected yearly fatal accidents rates per 100,000 hours for 2020-2039 that I am working with:

0.63
0.62
0.61
0.60
0.59
0.58
0.57
0.56
0.55
0.54
0.53
0.52
0.51
0.50
0.49
0.48
0.48
0.47
0.46
0.45

I am very confident in these yearly numbers, so I don’t need any assistance with them. Here is what I primarily need help with:

Even though these rates are technically from an exponential trend line curve, for simplicity I have been using the average of these which is around 0.54 with a reliability equation P = EXP(- #hrs * 0.54/100000) to get the probability of no fatal accident. In the end, I am turning this probability into a “1 in [1/(1-P)] chance” of a fatal accident occurring within the next 20 years vs hours flown. This all assumes number of hours flown per year remains constant. For 1500 hours flown for example, this equates to about a 1 in 124 chance.

Is this the best method to use to calculate the probability of a fatal accident occurring? I’m wondering if there are multiple methods for calculating this kind of probability instead of the EXP(-# hrs * rate) that may be better.

Anyway, if you can provide feedback or help with me determining the best method for this I would be greatly appreciative!

Thanks
 
What do you know of the Poisson Distribution?

Your 1:124 doesn't seem to be supported by your data. How did you calculate that?
 
tkhunny said:
What do you know of the Poisson Distribution?

Your 1:124 doesn't seem to be supported by your data. How did you calculate that?
Click to expand...

I'm sure I learned it back in school but honestly I don't recall much about it. However, after doing a quick search I think you end up getting essentially the same answer with Poisson and the reliability equation I used.

If I do both equations:
Poisson: ((t * lambda)r * exp(-t * lambda)) / r! = ((1500 * .54 / 100000)1 * exp(-1500 * .54 / 100000)) / 1! = 0.00803. 1/Ans = 124

The equation I've seen used elsewhere that I'm using: exp(-1500 * .54 / 100000) = 0.9919. 1/(1-Ans) = 124.

Since each of the 20 years are nearly linear in relationship, that's where I got the .54 / 100000 from. Is this what you are asking where I got my answer from or something else?
 
This requires a survival methodology. In order to have a fatal accident in year 10, one must first survive 9 years. In other words, your year-to-year probabilities are not independent. This is problematic.

From a practical point of view, 1/124 is horrendous. With the survival methodology, we see that we lose only about 1 pilot every 20 years, assuming they all fly 100 hrs / year. This is perhaps more reasonable.
 
tkhunny said:
This requires a survival methodology. In order to have a fatal accident in year 10, one must first survive 9 years. In other words, your year-to-year probabilities are not independent. This is problematic.
Click to expand...

My year-to-year numbers above are occurrence rates, not probabilities, so I'm not sure this concern is valid?

Unless do you have an issue with using individual yearly rates to calculate "surviving" across 20 years? If so, I'm still using a constant rate of 0.54. But if you want to ignore my 20 yr average of 0.54 and just use a different number, it would still come out to something close to 1 in 124. Using say 0.63 (my 2020 number) it comes out to 1 in 106. If this does not address your concern, please clarify.

I'm also failing to see how a "survival methodology" that you are referring to is any different in principle to what I'm doing. I'm essentially calculating the odds of someone not encountering an accident with a fatality ("surviving"). I use "surviving" in quotes because maybe the pilot survives but a passenger dies, but this is still considered a fatal accident. In the case of 1500 hours I'm seeing the probability of "surviving" at 99.2%.

To me, my calculation seems in line with something like a standard "what are the odds of no 1 in 100 year floods occurring within x years?". See:
It seems that I am essentially saying the same thing: "what are the odds of no 0.54 in 100,000 hours fatal accidents occurring within x hours?".

If you still feel I'm off here can you please provide a step by step calculation of this "suvival methodology" you are proposing to determine what you feel the probability should be?

tkhunny said:
From a practical point of view, 1/124 is horrendous. With the survival methodology, we see that we lose only about 1 pilot every 20 years, assuming they all fly 100 hrs / year. This is perhaps more reasonable.
Click to expand...

I agree that a 1 in 124 chance of a fatal accident for 1500 hours flying seems pretty high at first glance but I'm not so sure a number like this should be so surprising. Here is why:

The 2019 Nall Report (see "Source" below), has data through 2016, which is the latest year available. In 2016 for General Aviation (small airplane) Fixed-Wing (basically non-helicopter, so your standard airplane) Non-Commercial flying there were 1036 total accidents of which 159 involved fatalities. (There were 283 total fatalities). In 2013, 2014, 2015 there were 167, 196, and 197 fatal accidents respectively.

Given there are well over 100 fatal accidents each year, I'm not sure how we can come to this conclusion that losing only 1 pilot every 20 years given 100 hrs / yr is reasonable. It would seem that any methodology that came to this conclusions wouldn't be valid.

------

[Source: AOPA's Nall Report which according to itself in the 2018 report "has been accepted as the leading source for in-depth, factual reporting of general aviation accidents and accident trend analysis". The 2019 Report: https://www.aopa.org/training-and-s...oseph-t-nall-report/non-commercial-fixed-wing]
 
Last edited:
Can't argue with most of that. I may have had too narrow a definition of exactly who was flying and how often they did that.
 
tkhunny said:
Can't argue with most of that. I may have had too narrow a definition of exactly who was flying and how often they did that.
Click to expand...

Ok neat, thanks for reading my lengthy post :). I guess then I could consider my method to be a good one to use and the probabilities to be valid.

If someone has different feelings please let me know! Otherwise I'll assume it's probably good to move forward.
 
You must log in or register to reply here.
Top