A probability is one?

[Graeme M]

Sigh... My lack of formal education leaves me high and dry at times. Here's a really simple thing that I simply can't work out. I was reading a Scientific American article and this puzzle jumped out at me.

The author was observing that the chances of an event occurring change depending on whether the event has occurred or not. He wrote that he was issued a particular number plate and that the odds of this plate having the number/letter combination it did was one in 175,760,000 before he received the plate, but that "after the fact, the probability is one".

I don't understand this. I assume there is some particular view one takes when thinking about events statistically or probalistically, but my naive understanding doesn't suggest what that might be.

To my mind, the probability of an event can only be assigned prior to the event. Once it occurs, there is no probability at all, only certainty. Similarly, once the event occurs, its probability remains at whatever it was, in this case one in 175,760,000. That is, although it occurred, the probability of it occurring was what it was. Probability seems to me to be simply attempting to quantify the likelihood that an event might occur.

The only probability that might be assigned the value of one in one would be a certainty, for example the likelihood that a switch in an electronic circuit throws when the voltage exceeds a threshold value.

What am I missing here?

[NoShips]

I might have a heroic, though probably hopelessly incompetent, stab at this tomorrow, Graeme.

To begin with, though, there isn't just one interpretation of probability. There's epistemic (or subjective) probability, which we assign to events that are presumably entirely deterministic, due to limits of our knowledge. Then there's objective probability that has nothing to do with our cluelessness; certain events, we're told, are inherently indeterminate -- ask the physicists about quantum events.

As for car licence plates, I cannot possibly know what yours is, though I can assign it a subjective probability given what I know of the number of cars out there, the system of plate-naming, and so forth. After I see your Ferrari, I enjoy certainty. Objectively speaking, the plate number, we presume, was never in a Schrödinger's car type indeterminate state.

Nice to see ya back! Where have you been?

Edit: I ended up hogging your IT thread LOL. Sorry! You were elsewhere. Not my fault.

[NoShips]

The author was observing that the chances of an event occurring change depending on whether the event has occurred or not. He wrote that he was issued a particular number plate and that the odds of this plate having the number/letter combination it did was one in 175,760,000 before he received the plate, but that "after the fact, the probability is one".

So, subjectively speaking, the writer is correct (I think). The best he could assign prior to the event was 175,760,000:1. After his observation of said plate, he assigns a probability of 1. Objectively speaking, the number of the plate was determined by antecedent events. There was never any doubt to an omniscient observer, um, God or Donald Trump, say, to whom objective and subjective probabilities converge for deterministic events.

The writer is wrong insofar as he claims "the chances of an event occurring change depending on whether the event has occurred or not", assuming the event is deterministic, as the assignment of car licence plates is. With a more charitable eye, though, the righter is write insofar as our subjective assignment of the probability of said event changes with new information.

To my mind, the probability of an event can only be assigned prior to the event. Once it occurs, there is no probability at all, only certainty. Similarly, once the event occurs, its probability remains at whatever it was, in this case one in 175,760,000. That is, although it occurred, the probability of it occurring was what it was. Probability seems to me to be simply attempting to quantify the likelihood that an event might occur.

Subjectively speaking, yes. Objectively speaking... depends whether the event is deterministic (probability = 1) or not. Certainty is a state of mind, of which you being an eliminativist have none, tee hee, nature will do what it must do. Prior to the occurrence of a determined event, you do the best you can in assigning the chances of it happening. To the determinist, that you would post what you did today was pre-ordained; it had an objective probability of one. None of us could know this though. Personally, it was a complete surprise. I would've assigned it a subjective probability of 0.0001. Now I assign it 1.

Had a few beers LOL. Don't trust a word. Great topic though.

[Braininvat]

It's just semantic choices. Probability doesn't change for getting a plate, but our expectation of it changes after we get that plate. If there were a bin full of baby souls, randomly distributed to planets, then my chance of being born on Earth is .0000012, say. When I am born, the chance is 1.0 because I can't reasonably expect to be anywhere else. The chance for the other bin-babes remains at .0000012. As NS said, it's about new information.

[NoShips]

It's just semantic choices. Probability doesn't change for getting a plate, but our expectation of it changes after we get that plate. If there were a bin full of baby souls, randomly distributed to planets, then my chance of being born on Earth is .0000012, say. When I am born, the chance is 1.0 because I can't reasonably expect to be anywhere else. The chance for the other bin-babes remains at .0000012. As NS said, it's about new information.

Noooooo!!! I said nooooo!!!!!

That suggests the dude who says the probability is 900 million to one (or whatever it was) is saying the same thing as the dude who says the probability is one. Don't give semantics a bad name LOL. Equivocation, your honor!

The rest sounds brilliant as ever, BiV. :-)

[NoShips]

"Hi, I'm here in response to your ad for a semanticist"

"Sorry, you guys are fulla crap. You all say the same thing. Next!"

Tee hee!

Edit: "Woe is me. Where to go from here? Hey wait, I know..."

:-)

[someguy1]

To my mind, the probability of an event can only be assigned prior to the event. Once it occurs, there is no probability at all, only certainty.

Consider the following thought experiment. I flip a coin and show the result to A, and hide the result from B. I ask A and B what is the probability that the coin landed on heads.

A is rational to say either 0 or 1. B is rational to say the probability is 1/2.

The point being that probability is not anything inherent in the event itself. Rather, probability is a measure of our state of knowledge of the event.

Under highly controlled laboratory conditions, coin flips are deterministic. How could they not be? You start with the coin in a particular orientation and location in space. You impart a given force. The air resistance is such and so. It would violate the laws of physics for the flip of a coin to be truly random.

Yet, when we flip a coin, the odds are 50-50. Why? Because we are ignorant of the exact position, the exact force, the exact air pressure.

Probability is not inherent in the event. Probability is a measure of our state of knowledge.

Here is yet another real life example. I go to the airport. The authorities give me a full body cavity search. They examine the contents of my luggage. They take my laptop apart down to the screws and chips.

What is the probability that the flight has been made safer?

To the security authorities, it's negligible but nevertheless nonzero. Anyone could be a terrorist and they don't know who is and who isn't, and they just eliminated me from suspicion.

To me, the probability is zero. Because I know I'm not a terrorist.

Once again we see that probability is a measure of subjective knowledge.

[NoShips]

Here is yet another real life example. I go to the airport. The authorities give me a full body cavity search. They examine the contents of my luggage. They take my laptop apart down to the screws and chips.

What is the probability that the flight has been made safer?

More to the point, what are the odds they discover the bottle of Johnny Walker stuffed deep in your GI tract?

[Braininvat]

I have a clever way of hiding JW - I take it out of the bottle first, then hide it in my GI tract.

Someguy: becoming a Big Fan of your lucid style of explanation. I was getting at the same thing, but less clearly.

[Graeme M]

The point being that probability is not anything inherent in the event itself. Rather, probability is a measure of our state of knowledge of the event.

Yes, you see that's exactly what confuses me. That "definition" is precisely what I'd have naively expected a quantified probability to mean. It is simply a measure of our uncertainty about a future event. But once an event occurs, there is no longer any uncertainty and hence no probability. To my mind we cannot talk about the "probability" of an event that has occurred. Our state of knowledge after an event is a different thing, so it seems to me!

I think this just means some kind of semantic thing on my part - I just don't see how we can talk of a "probability" for a past event, except in terms of what the probability was prior to its happening.

But you are saying that we can still talk of the probability of an event after the fact, it's just that now we know that the event has occurred and hence its probability is one. Fair enough.

Still doesn't sound reasonable to me though! :)

[NoShips]

The point being that probability is not anything inherent in the event itself. Rather, probability is a measure of our state of knowledge of the event.

Yes, you see that's exactly what confuses me. That "definition" is precisely what I'd have naively expected a quantified probability to mean. It is simply a measure of our uncertainty about a future event. But once an event occurs, there is no longer any uncertainty and hence no probability. To my mind we cannot talk about the "probability" of an event that has occurred. Our state of knowledge after an event is a different thing, so it seems to me!

I think this just means some kind of semantic thing on my part - I just don't see how we can talk of a "probability" for a past event, except in terms of what the probability was prior to its happening.

But you are saying that we can still talk of the probability of an event after the fact, it's just that now we know that the event has occurred and hence its probability is one. Fair enough.

Still doesn't sound reasonable to me though! :)

You're talking subjective probability, i.e. degrees of belief (which varies from person to person, and time to time, depending on available info, degree of rationality, and number of Pilsner Urquells consumed)

The objective probability of inherently indeterministic events (quantum stuff, say) does not change. The chances of said uranium atom decaying in a certain timeframe was 0.362 (or whatever) - and always will be!

[Graeme M]

Well yes, I think the article was simply discussing probability in a deterministic sense, rather than touching on quantum indeterminacy. The author was discussing statistical probability and "survivor bias" whilst reviewing a book called "Standard Deviations".

In the book another example is given:

The odds of me having a particular playing hand in a game of poker is about three million to one, but the book's author states that "After I look at the cards, the probability of having these cards is 1, not 1 in 3 million".

I disagree. The probability that I have that hand remains the same, the fact that I do have them is now in evidence. Probability must surely be a measure of ignorance, not knowledge.

[someguy1]

Someguy: becoming a Big Fan of your lucid style of explanation.

Oh to be as persuasive on the political forums :-)

[NoShips]

Well yes, I think the article was simply discussing probability in a deterministic sense, rather than touching on quantum indeterminacy. The author was discussing statistical probability and "survivor bias" whilst reviewing a book called "Standard Deviations".

Cool. From now on we'll regard probabilities as subjective; a number between 0 and 1 that we assign to deterministic events the occurrence of which we are uncertain.

In the book another example is given:

The odds of me having a particular playing hand in a game of poker is about three million to one, but the book's author states that "After I look at the cards, the probability of having these cards is 1, not 1 in 3 million".

Deferring to the cleverness of mathematicians, it is indeed the case that the probability those who are ignorant of your royal flush, or pure crap, or any other hand, is three million to one. After all, I don't have X-ray specs. The probability you assign to your having a royal flush, assuming you have one, is one, assuming also you're not a Cartesian skeptic and trust your eyes.

So far, then, just after the cards are dealt, I assign a probability of 3,000,000:1 that you have any given hand (till you grin like a clown, of course, then I adjust); you assign the probability that you have the hand you were dealt 1, on the grounds you can see it.

I disagree. The probability that I have that hand remains the same, the fact that I do have them is now in evidence. Probability must surely be a measure of ignorance, not knowledge.

It remains the same to you (assuming you peeked). It was one to begin with, upon looking, and still one. To me, it was horribly unlikely to begin with; now, calling your bluff, I assign it 1 too (or in layman's terms: "That fooker has a royal flush"), and sportingly throw over the card table and shoot you.