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Next Time Somebody Asks You “What’s the Probability of X” Say This — Crucial Update!

Next time somebody asks you “What’s the Probability of X”, where X is any proposition, say this: there is no such probability. It doesn’t exist.

“Briggs, what’s the probability of this die coming up 6?”

It doesn’t exist.

“Briggs, and am deep sixed?”

I don’t know: there is no such probability.

This is your periodic reminder that no probability exists unconditionally. You can never have a probability without reference to some evidence, premises, supposeds, model, whatever.

That is, you must never write

     Pr(X)

but you should write

     Pr(X|E)

where E is the evidence brought to the problem. Change E, change the probability of X with respect to E.

Now you will see textbooks write probability the first way. Usually this is shorthand, because authors are lazy or the text appears busy when writing it properly. Do not let this fool you. Or sometimes the authors believe such probabilities exist. Frequentists believe this. They are mistaken.

You can conceive of no probability without reference to some evidence—and that evidence always includes the knowledge of the words and grammar of the proposition X. That evidence is implicitly in E, even if you fail to write it down.

This should not come as a surprise, since no proposition (in logic) is either true or false without references to the evidence used to judge is true or false. The evidence used to decide truth or falsity always includes the word knowledge and grammar, too. I emphasize this, because it often forgotten.

Now in real life if you ask me X = “What is the probability I die?” I will say it is certain. This is because I am using shorthand, as are you. The E with which we are using to judge the proposition is tacitly understood and agreed to by both of us. There is no reason to belabor or elaborate something so obvious.

This commonality is present in most mundane probability questions. But it’s only there because of shared cultural experience. If there is any doubt, the premises E must be set down explicitly.

This is why (as we have often discussed) there is no probability of being struck by lightning, or having a car accident, or even of winning the lottery. No probability exists. Probability is always a deduction from accepted evidence.

That includes all scientific probabilities, too. Including all the medical trials reported in the news. Nobody has a probability of cancer, or even of health. The reason is there are no such things, because there are no referents. Always insist on one!

Update I’m elevating a comment to the main post, because it highlights a common and devastating error.

A commenter below wrote “If P is a proposition, then the proposition P AND ~P is false, always, not matter the ‘evidence’ or anything else.”

This is false, and easily seen to be false as long as you keep in mind the cautions above. Every proposition that is evaluated is conditional on stated and unstated or implicit conditions, some of which include the word or symbol definition and grammar. This cannot be stressed highly enough, for even after it is said, it is forgotten, as this commenter proves.

For instance, is this proposition (enclosed in quotes) true or false?

    “♠ ⌈ ‰”

You can’t tell. (The probability it is true given these unknown symbols, as proven in this award-eligible peer-reviewed book is “(0, 1)”.) Why? Because you have no idea what the symbols mean, nor how to manipulate them (the grammar).

But if I tell you ♠ = P, where P is a proposition, and ⌈ = AND (logical) and ‰ = not-P (but only when preceded by ♠), then we are back to “P AND ~P”, which we still can’t say is true or false until we recognize the implicit unspoken assumed premises; i.e.

    Pr(P AND ~P | what I know about logic, the symbols P, etc.) = 0.

Proving, as claimed, that no proposition stands alone. Just because you’re not writing the right hand side down, doesn’t mean it isn’t there. Recall, too, probability is a matter of epistemology, and not causality. We are not saying why any proposition is true, but how we know it is.

19 thoughts on “Next Time Somebody Asks You “What’s the Probability of X” Say This — Crucial Update! Leave a comment

  1. “Nobody has a probability of cancer” Of all the probabilities of my health outcomes, losing part of my tongue and having a great “knife fight” scar on my neck was never even on the possibility chart (though everything technically is possible). There were so many other things. All the evidence said “Nope, won’t happen” yet there it was. Even adding the evidence can give false probabilities. Which is why avoiding the idea of probability might be a better way to live life. Cuts the false worries and dread of outcomes that never happen and allows one to deal with the improbable that does. Knocks out that “fairness” nonsense, too.

    Yeah, we know driving 100 mph on the freeway can kill you and others. It’s probable. But it’s also common sense. Maybe if we just used common sense and left out the “math is soooo hard” part, we’d be better off.

  2. “Why is there something rather than nothing”? Is a good question.
    Then, once there’s something, talk of probability given what is known, or history, or normal sequences of events, or ‘evidence’, comes in. That’s how I understand it.

    Sheri, sorry to hear about your history.
    Maths hard or easy, is irrelevant in such simple discussions. Only if someone’s trying to make something seem exclusive in some way.
    The point Briggs makes in this post he make in his Breaking The Law Of Averages book. An easy read!
    I stopped when the computing started…chapter five!

    Binomial, differential equations, are easy at school, about 12 or 13, Not when you’re doing them in your lunch break at work under a CCTV from instructions that are yankie who can’t even spell Maths. At which point you realise your brain has maybe changed, for good, maybe even for the better, who knows.

  3. “The E with which we are using to judge the proposition is tacitly understood and agreed to by both of us. There is no reason to belabor or elaborate something so obvious.”

    Well, that sure seems reasonable, and, rebuts the earlier remark made:
    “”… what’s the probability of this die coming up 6?” … It doesn’t exist.”

    I appreciate Briggs’ rigor (except for some of the semantics nitpicking). Really. But I can’t help but notice a fundamental inconsistency:

    Also said: “You can conceive of no probability without reference to some EVIDENCE — and that evidence always includes the knowledge of the words and grammar of the proposition X.” [EMPHASIS added]

    Hhhmmmm. Consider the various remarks made a [not here today, but here on other days], where essentially if not precisely the absence of evidence serves as the basis for reaching a particular conclusion: It’s a miracle! Where science might present an alternative, more hum-drum, explanation science gets belittled. Consistently.

    As Feynman pointed out, science might prove many things, but fundamentally what science can prove is when some specific thing is false: http://www.youtube.com/watch?v=EYPapE-3FRw (see about 3:50 – 5:00) Feynman also points out that vague theories cannot be falsified (about 5:30), making their value limited for furthering knowledge, but perhaps useful as a basis for seeking additional information — what Briggs refers to here today as “E” (see Feynman starting at about 7:30 about why guessing a good hypothesis is important).

    Isn’t it curious that “science” endeavors to do the very thing endorsed as mandatory — refining and finding the “E” using today’s essay’s jargon — and at the same time is also a recurring target for belittlement … when all science can do with certainty is falsify [or define limits for] a hypothesis or belief…..

  4. Ken,

    The first statement does not rebut the second one you cite. The probability of a die landing on 6 is not a priori 1/6. There is a condition, E, along the lines of that it is a fair die, or perfectly cubical, or the like.

  5. “no proposition (in logic) is either true or false without references to the evidence used to judge is true or false”

    If P is a proposition, then the proposition P AND ~P is false, always, not matter the “evidence” or anything else. Also, P OR ~P is always true, under any set of conditions. Those are just a couple of examples, both of basic logic facts and of this author’s utter confusion.

  6. You did indeed demonstrate something, but not what you intended. Anyone who has mastered the first chapter of any textbook on symbolic logic knows what the score is.

  7. Lee,

    Your comment is that of a petulant student, unwilling (through pride? inability? incomprehension?) to answer the argument itself.

  8. Lee – you make the classic blunder of getting involved in a land war in Asia. But only slightly behind that is assuming that P OR ~P is logical or even appropriate. It’s so easy to forget that the answers are not always true or false. Or that there are always answers.

    The real world works on true, false, sometimes, maybe, not addressed, {out of cheese error redo from start}, “Whachoo talkin bout Willis?” The real world is not an equation of symbolic logic, but a magic 8-ball of the known, unknown, and unknowable.

  9. That a statement of any opinion or fact, is always known to be based on something: We have the word ‘because’.
    Where that something might be word of mouth, i.e. media, bad information, (rubbish), or some rigorous study into a matter, is never explored if the opinion is disliked. The projection and cliche’d personal attacks start prior to exploring the thinking.

    It is always assumed, if the opinion is considered “wrong” that the individual must have no evidence, or questionable evidence, by default, since the opinion is opposing.
    This is so often mischaracterised as relativism. It is not:

    Sometimes, people’s milage varies, to take Michael 2’s phrase.

    Some lack experience and believe their superior calculating ability is sufficient. Some think they are,
    “Ahem, on the side of the angels” I quote directly,
    So they can get away with, and do, anything they like with impunity.
    THAT is relativism.

    So, when the commentator is critically biased, dissecting the opponent’s argument, extreme assumption and mean logical literalism prevails. When the argument offered is on your side, all is forgiven, and ‘tacit givens’ are accepted and little points are ignored.

    The generous and honest commentators don’t nitpick unless they’re working on editing a piece of work, maybe appealing for greater accuracy or standards. Which is why I like audio. There is little room for crappy, cheap, runaway shots. Audio that’s like that falls flat.

    Thank you for the Feynman lecture, Ken, I look forward to hearing it.

  10. If P is a proposition, then the proposition P AND ~P is false, always, not matter the “evidence” or anything else.
    Among the evidences are the meanings of ‘proposition’, of ‘and’, of ‘false’, ‘always’. The rules of the sentential algebra, etc.

    Not to mention, the meaning of the term ‘evidence.’

  11. Death threats, or statements of the bleeding obvious?
    die, of course we mean dice, sixed, deep six *see urban dictionary; or 6/10 drunk,
    “Briggs and am” . Meaning? Does that matter, semantically?
    Chances of being knocked down by a car?
    “being pushed down the stairs”
    “run down by a taxi driver” (real life interpretation)?
    “beamed in the head’?
    “winning the lottery”

    and on and on.
    “Falling off a roof?” is that roof in the sense of the word REEF? I’ve had to phone friends about it.
    “Being buried under a canoe”
    “strangling with bedsheets”

    https://www.youtube.com/watch?v=mNbSgMEZ_Tw

    It makes no sense either, but some gifts are more than just a gift.

  12. You (plural) keep using that word. I do not think it means what you think it means.

    Most people think ‘evidence’ means ‘proof.’ It doesn’t. cf. https://www.manhattancontrarian.com/blog/2018-11-11-what-does-it-mean-that-there-is-no-evidence for a lawyerly view.

    But then Briggs, in the OP stated:
    You can never have a probability without reference to some evidence, premises, supposeds, model, whatever.

    And then Chastek comments briefly in another context:
    In speaking about energy we wave our hand at “the ability to do work”, and then grab some equation for work. But was there really nothing to notice in “Ability”?
    IOW, “ability” was one of those unspoken premises that were assumed in an informal way and so went unnoticed; just as the nature of the relevant modes of logic were assumed in stating that “P and NP” was necessaily false.
    https://thomism.wordpress.com/2018/11/16/the-informal-strikes-back/

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