The *National Journal* says: “The odds of being killed by a shark are about 1 in 3.7 million. The odds of being killed by a sting from a bee, wasp, or hornet are 1 in 79,842, according to the National Center for Health Statistics, a part of the Centers for Disease Control and Prevention.”

Now you see these kinds of stories all the time, all of which have conclusions like You have a better chance of dying in a lightning strike than in winning the lottery, etc., etc.

These stories are all wet. Nobody has a chance of dying in a lightning strike, just as nobody has a chance of winning the lottery or being killed by stinging sharks or biting wasps. By which I mean, everybody has a chance of winning the lottery or being stung by lightning or whatever.

Double talk? The problem is *chance*, a nebulous word, apt to change shape mid-sentence so that you don’t always end up where you were aiming.

In one version, *chance* means *logical possibility*. Everybody has the *logical possibility* of dying by shark, bee, or lightning. But *chance* also connotes *probability*. And there just is *no* unconditional *probability* of dying by anything.

Logical possibility is a weak criterion. Anything that isn’t logically impossible, such as square circles, is logically possible. You might be a native Antarctican, solid ice your bed and fluffy snow your pillow since birth. But, one day, an evil Polar Vortex might surreptitiously search the seas for your doom, and fling a hammerhead shark from the tropics to the very spot on which you take your morning ice floe stroll, as was illustrated in the documentary *Sharknado*.

Hey, it *could* happen. It’s logically possible. And in that sense you have a *chance* of dying by shark bite.

But you don’t have an unconditional probability of dying by one. What can we say? If the probability of you being eaten by a shark were 0, then it would be impossible, logically impossible, that you could be eaten by a shark. But we’ve already agreed that it *is* logically possible. And if your probability is 1, then that means the universe guarantees, no matter what, you will have your head bit off. Ouch.

This means the probability of you dying by shark bite, without knowing anything else about you (and I mean this clause just as it’s written), is no number at all, but all numbers between 0 and 1. Which is fairly useless, as far as information goes, But not entirely unless since non-extreme probability tells us a event is *contingent*, i.e. logically possible.

We can now see that it makes no sense to say the unconditional probability of dying by a bee sting is “larger” than of suffering the consequences of a sharknado. These probabilities, in the absence of any other information, are equal.

What “other information”? Example: your aunt Narantsetseg lives in central Mongolia, far from the sea and inland aquariums, whilst you live in Key Largo in a beach shack. Given only *this* information, which includes common knowledge about these locations and their nearness to sharks, it’s natural to say you have a larger probability of dying of shark bite than your aunt.

How much larger? We don’t know. There’s still not enough information to quantify the difference. Why? *All* probability is conditional on the information supplied. If that information is vague, as it is in the maximal sense when we know noting other than the event in logically possible, no quantification is possible. To get numbers, we need specific information.

Enter the the Frequentist Fallacy. Happens like this. American citizens killed by shark bites are divided by the population, and this number is substituted for the probability of you yourself dying from shark bite. This “probability” is assigned to beach dwellers and Norther Michiganders alike. Which is silly, because, obviously, the information for these folks is radically different, and thus so are their (conditional) probabilities.

So is dying of a bee or wasp sting more probable than by shark bite? We now see that it makes no sense to ask this. If you can’t swim and are allergic to bee stings and live next to an apiary, then it’s more likely you’ll die from a sting than a shark bite. But if you live in Key West and go snorkeling daily and aren’t allergic to bees, it’s more likely you’ll die inside the innards of a shark.

How much more likely (in either case) we can’t say.

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*Thanks to Brad Tittle for suggesting this topic.*

Categories: Statistics

I don’t see any frequentist fallacy there. Your post is related to the problem of assigning probabilities to a single cases. That is a problem for which some very sensible remedies (e.g. W.C.Salmon’s) exist i.e. estimate probabilities for the largest homogeneous reference class possible. Furthermore yes we can talk about probabilities – those assigned to classes but not single cases – even when this condition is violated. Say you’ve been asked by an international organisation (e.g. WHO) to provide three kinds of life insurance – for mosquito/bee sting/shark caused deaths, – to the whole human population. Would you charge the same premiums (or/and keep same reserves) for all types of insurance?

This was an interesting example of probability gone wrong:

“It’s a 50-50 bet that the thin Arctic sea ice, which was frozen in autumn, will completely melt away at the geographic North Pole, Serreze said.” (Mark Serreze. artic researcher). One would hope that Mr. Serreze meant that based on the factors currently at hand, combined with much research, there is an equal chance the ice will melt completely as that it will not. However, most people, perhaps even Mr. Serreze, will say something like : “Of course. Either it will or it won’t”.

I don’t know if the speaker realized that his “prediction” is just going to look silly to many people because of his phrasing, or not. However, stating a 50-50 probability with no qualifications or other information may lead people to believe you’re saying it either will or won’t happen. Not useful.

It’s a very common statement on probability and mostly used incorrectly.

What we are seeing is a rate , that is, X/N people die from C (including the big-C) over some period T and that is being called a probability. It’s not of course.

Speaking of which, are lightning strikes (of humans) on the rise? I hear about them more than before but that might be because The O being a Do-Nothing president is no longer news. OTOH, there are more people (in unprecedented numbers, no less) being spread over a larger area.

DAV: Couldn’t find any current stats on lightening strikes, so can’t really say. I’m sure they are reported more because that’s Extreme Weather and therefore proof of AGW. Or so it’s claimed by the media. (And it does probably have to do with slow news times.)

Lightening strikes depend on the number of thunderstorms (which is pretty high this yearâ€”but not extreme) and how many people are outside at the time. I know in national parks, there are people struck by lightening because they ignore the signs that say no hiking in the afternoon due to lightening. Okay, maybe not everytimeâ€¦..Anyway, since more people are outdoors and hiking in more open areas, there are more people struck by lightening. Have you noticed that is often not fatal?

All probability is conditional on the information supplied. If that information is vague, as it is in the maximal sense when we know noting other than the event in logically possible, no quantification is possible.This statement is eerily similar to Aquinas’ demonstration that the will is free, since that also depends on the extent of one’s knowledge.

Eppy,

The Frequentist Fallacy better put is assuming relative freqencies are probabilities. They never are. But, of course, knowledge of rel. freqs. can inform probability.

YOS,

And somehow you remind me I haven’t yet bought this book by Jim Franklin: http://www.amazon.com/Aristotelian-Realist-Philosophy-Mathematics-Structure/dp/1137400722/

But look at the price! Youch!

The probability that a person chosen randomly from a population (i.e. a simple random sample of size 1) is killed by a bee equal to Kb/N, and a probability of being killed by a shark equal to Ks/N. Therefore, the probability is higher to be killed by a bee because more people are killed by bees than by sharks, and there is no fallacy here. Saying “you” instead of “a randomly chosen person” is a colloquial way of talking that most people understand properly. Furthermore, it gives real, valuable information about the world.

If I say “your probability of being a male is about 50%” you will immediately complain that that’s a fallacy, either saying that it is 1, or that it is 0. But if you make the little effort to interpret what I say, you will see there’s no fallacy, but information from the real world about the composition of the population.

Furthermore, if for you unconditional probabilities do not exist /make sense/ are fallacies, how much information should we pour in so that the resulting conditional probability ceases to be a fallacy? No added information will ever satisfy you, and you will end up negating every probability, unconditional or not, as a fallacy.

Let’s not be so hypercritical and use some common sense to interpret what is being said. Yes, it is more probable that “you” die in a car crash than in an expedition to alpha-centaurus.

Felipe,

Dude, no. Point is that it is your understanding of probability which is flawed. Starting with your committing the frequentist fallacy right out out of the gate.

My point was far from being hypercritical. It was instead decently hopeful that if we did things the right way, our understanding would increase.

Currently, it is more probable that “you” die in a car crash than in an expedition to alpha-cenataurus, since the latter does not currently exist. While not logically impossible, it is impossible at this particular time.

Sheri,

You missed the point, the problem is not whether or not car crash or expectations to Alpha-Centaurus exist, the problem is who is “you”. If “you” live someplace where there are no cars, both are impossible for that “you” at this time. Now if you want to talk about the average “you” yes you statement might be true, except there is no “average you” and the “average you” is only a construct that allow small minds to try to understand risk on only the simplest terms which most of the time end up wholly incorrect for the risk to you a person who live is a specific place and time and who make decisions based o that place or time which cannot be average out over a large population or area.

Epanechnikov as to life insurance example you miss understand what an insurance companies actuary table accomplish, it not about risk to the “you” it is about the risk to the company for the pay outs to the “average you” they could care lees what happens to you but what happens to the “average you” which again does not exist and is only a construct which serve r the insurance company very well but for you not so much other than you are paying for the cost for the “average you’ and depending on your behavior may or may not be a good deal for you.

Briggs please correct me if I am wrong.

Have you noticed that is often not fatal?Yeah and that’s surprising. The induced strikes at the lightning strike lab in Florida fuse beach sand into glass and direct hits on frame houses with brick facades does extensive damage including removing most of the facade. I assume most of the hits reported with survival are not direct hits but smaller eddy and secondary currents.

DAV: Yes, most lightening strikes on humans are not direct hits. Some people are even “blessed” with being hit twice. It’s a pretty interesting phenomena (except to those are hit, one would think).

Mark: Okay, if “you” are in a place where there are no cars and there’s no possibility of cars being anywhere near you and you never travel to a place with cars, then both examples are impossible. I think that there is an assumption that the “you” applies to the society where the example is being given, thought there really is no reason that has to be true.

Mark Luhman as for the life insurance I thought I was clear by stating that we can talk about probabilities “assigned to classes” even when we can’t say anything about the single case. My point is that if you were to insure the whole (yes heterogeneous) population it would have made perfect sense to take into consideration those frequencies. Furthermore insurance companies are not there to insure the “average you” – they are there to cover a sum of claims. Modelling the first moment could be useful but the total risk can’t be estimated solely by studying the behaviour of an average. Quite frequently (e.g. fat tailed distributions) the mean doesn’t tell you much at all of how possible it is to run high loses.

There’s lightning (not “lightening”, Sheri. That’s something quite different) in Missouri and North Dakota as I type. I find http://www.lightningmaps.org/realtime?lang=en totally fascinating. Lots in the sea between Florida and Bermuda too. Probably hitting sharks.

Briggs,

I see you are using the term “Frequentist fallacy” in the De Finetti’s sense. I agree with you that relative frequencies are not probabilities (their limit however might be but then there is no definite justification on that – Hans Reichenbach’s attempts were indeed very sensible but not entirely satisfactory) but they can be useful in estimating them. However that is equally true whether we consider unconditional or conditional probabilities and there is no probability interpretation which totally/logically succeeds into assigning probabilities to single cases.

Epanechnikov and Sheri It seem we are not on the same page, you want to talk about about the group we belong to and what I get from above Briggs is discussing the risk to “you”, the unique you, and how the group classification does not serve the “you” well, which I agree with. Since you seem to not to be able to grasp the concept of “you” and not the group, I see little value in you thoughts on the subject since you are unable to leave the group behind and concentrate on how little value group classification serve and individual, yes the above statistics stated have value to the group but not the individual. In my case the above are of little value, first I do not swim in oceans and I do have anaphylactic reactions to insect stings, the above statistics to me the “you” are worthless. So in my case Briggs arguments do make sense.

Mark Luhma could you please read what I’ve written above:

“there is no probability interpretation which totally/logically succeeds into assigning probabilities to single cases”

That is true even when we consider conditional probabilities! That is because we never deal with completely homogeneous groups and we lack the absolute knowledge which would have allowed as to condition enough number of times!

Hence, as I initially stated, the Briggs’ post is more closely related to the problem of assigning probabilities to single cases that it is to the frequentist fallacy .

p.s. excuse my numerous typos

Rich: Noted on the spelling. Very different.

Mark: I don’t have time to explain and “you” probably wouldn’t understand anyway, so I am signing off.