Wikipedia:Reference desk/Archives/Mathematics/2021 October 29

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October 29

Odds, probability...

I was trying to explain to my kids how much safer flying is than driving, and I found this statistic online: "It is estimated that the odds of dying in a plane crash are 1 in 9,821. For a better perspective, that is 1 fatal accident per 16 million flights. Whereas the odds of dying in a car crash are approximately 1 in 114." So my astute 16yo daughter asked me what that even means...and I had to admit that while I'm a science guy, I'm not a math guy, and I don't really know.

Obviously, we discussed, it can't mean that 1 out of every 114 car rides will result in a death, or else everyone in the US would die, on average, about 3 times per year (being that we use cars so often and so 114 x 3 = 342 car rides per year is something almost everyone experiences). And also, 1 in 9,821 is seemingly explained as 1 death in 16M flights, with the obvious issue that since planes carry so many more people than 1, the clustering effect would make it that no one person can die from a plane crash...but I didn't know how to figure that in. So I told her that odds are different than probability, but I didn't really know how to figure out the math. So I was hoping someone could give me a quick explanation so that I can understand it better and I can explain it to my kids better. Thanks! DRosenbach ^{(Talk | Contribs)} 12:43, 29 October 2021 (UTC)

@DRosenbach: it appears that the "1 in 114" (or 1 in 103, or 1 in 107 depending on the source) figure is the lifetime chance that you will die in a car crash (so 1 in 114 people die due to car crashes, not 1 in 114 car crashes cause deaths). I assume the 1 in 9,821 figure is working the same, but it wouldn't really give a direct comparison as people drive much more frequently than they fly. I'm not really sure if it's possible to find a direct comparison to the 1 in 16 million for driving, but based off of an online search it seems that in the US there are 1.20 deaths per 100,000,000 mi (160,000,000 km) driven. Based off of the probabilities you'd expect driving any more than 5.2 mi (8.4 km) to be more dangerous than flying, though of course there are tons of other factors such as the type of road, the length of the flight, etc. Hope this helps. eviolite (talk) 13:20, 29 October 2021 (UTC)

(edit conflict) A link to the source of the statistic might be helpful. I would guess that the figures given are as proportion of all deaths, that is: one out of every 9821 humans/Americans/... dies from airplane accidents, one out of every 114 humans/Americans/... dies from car accidents. (This roughly matches the data from List_of_causes_of_death_by_rate#Detailed_table for car accidents, and planes might be filled under "other transport injuries".)

This does not immediately say whether flying is safer than driving. After all, most people drive more than they fly, so that could explain why flying casualties are lower. Many activities are incredibly dangerous but kill few people precisely because they are known to be dangerous (obligatory XKCD about the risk of being struck by thunder). You would probably need to look at the death risk per travel or per distance travelled to have a more informed view. Tigraan^{Click here for my talk page ("private" contact)} 13:24, 29 October 2021 (UTC)

Resolved

OK wow...great points made by both responders. I just knew I could count on you guys at the math reference desk!! (and that's not just a retrospective analysis, counting the hits and not the misses :) ) DRosenbach ^{(Talk | Contribs)} 14:19, 29 October 2021 (UTC)

The risk of dying from jumping off a tall building is close to 1 in 1 – but only for people who jump off buildings. The lifetime chance of dying in a car crash is highly dependent on the population, and how it is sampled. The lifetime odds of a Nicaraguan campesino dying in a plane crash are closer to 1 in a gazillion; their risk of being hit while at work in the field by a crashing plane may be higher. A more meaningful measure (which is, however, age-dependent) is by how much one's expected lifetime is reduced by some action like taking a car trip – which does not strongly depend on how often one travels by car. Note that "Motor vehicle road injuries: 5.4 to 6.0" (per 100,000), as listed in List of causes of death by rate, is several orders of magnitude lower than 1 in 114.  --Lambiam 15:24, 29 October 2021 (UTC)

This is, like many real-life related math/statistics questions, nuanced and dependent on how you define and collect the data. For example you can easily obtain statistics on the number of deaths to air-travel accidents: This US government site puts the value at 2 deaths per 100,000 hours of flight. This related US government site puts the number of motor vehicle deaths at about 1 per 100 million miles. Unfortunately it's difficult compare hours of flight to passenger miles. One question that needs to clarified is whether "safer" means fewer injuries/fatalities per mile traveled or per hour traveling. This would make a big difference since I estimate airplanes average about 10 times the speed of motor vehicles. But what are the other risks of being in an airplane? You are exposed to more radiation, you may be exposed to diseases that you wouldn't come into contact with in a car, and if you have a medical emergency you may be unable to get timely life-saving treatment. You can ask similar questions about the risks of being in a car. There are less tangible factors as well, for example what was the relative effect of air travel vs. motor vehicle travel in the spread of COVID? The popular media seems to be very good at obfuscating the numbers even more since they often don't pay attention to relevant definitions when reporting statistics. The upshot is that this question involves a lot more than just math and I'm guessing that you'd need experts in relevant specialties to get a reliable answer. --RDBury (talk) 20:06, 29 October 2021 (UTC)

If the discussion is about safety and not only death then you should also consider the risk of non-lethal injuries. The death to injury ratio is high in planes and low in cars. That means the safety advantage of planes is larger than if you only consider deaths. But people scared of flying may just hear "We're all dead if it crashes". PrimeHunter (talk) 22:56, 29 October 2021 (UTC)

That's a good point. It's difficult to interpret the information available from the NTSB, but aviation accidents seem to range from relatively minor (damage to the aircraft but no injuries, to 100% fatal. Actually fatal aviation accidents are relatively rare in the U.S., with 3 fatalities in the last 10 years compared with 40 serious injuries. (I getting this from the 2001-2020 Preliminary Aviation Statistics​​​​ spreadsheet available here. The definitions of "fatality" and "serious injury" aren't really clear from the spreadsheet though.) The last really serious accident in the U.S. seems to be Colgan Air Flight 3407 in 2009. --RDBury (talk) 01:54, 30 October 2021 (UTC)

I wonder if the relative risks have changed due to the pandemic. Even if there's no more plane crashes than before, there is now more violence aboard planes, and covid infections spreading in planes and in airports. 2601:648:8202:350:0:0:0:D4A (talk) 00:50, 31 October 2021 (UTC)