When is it rational to believe in the improbable?
When someone shuffles a deck of cards and deals you the first twenty cards, the probability of getting those specific cards is extremely unlikely. Yet we have no problem accepting that you will get an extremely unlikely hand.
On the other hand, it's also extremely unlikely that your child will ever be a member of the National Basketball Association. Almost no one will accept that their son will be in the NBA. It would be considered foolish to believe.
What is the fundamental difference between these two examples? And is there a principle on deciding whether or not it is rational to accept the improbable?
On the other hand, it's also extremely unlikely that your child will ever be a member of the National Basketball Association. Almost no one will accept that their son will be in the NBA. It would be considered foolish to believe.
What is the fundamental difference between these two examples? And is there a principle on deciding whether or not it is rational to accept the improbable?
Comments (34)
If K is your entire body of knowledge at the moment, and E is some hypothetical event, then E is not believable if the conditional probability of E given K is low:
P(E | K) << 1
In your first example, the probability of a specific sequence of cards being dealt from a deck (E[sub]i[/sub]), given that the deck is shuffled and the dealer is not cheating (K), is low:
P(E[sub]i[/sub] | K) << 1
But the probability of any sequence being dealt is a certainty:
P(E[sub]1[/sub] V E[sub]2[/sub] V ... V E[sub]N[/sub] | K) = 1
So if this case seems puzzling, that just means that you were confused about which event you were considering.
Likewise, in your second example if the event is that a specific child ends up in the NBA, the probability is low, but the probability that someone's child ends up in the NBA (given that NBA is still around by that time) is a certainty.
I think we’d call that statistically ‘impossible’ rather than ‘extremely unlikely’. The definition is where it’s not worth a mention.
Where that line is is pretty much a subjective judgement clouded by hopes and fears.
Never.
Being rational is using reason and using reason is providing reasons to support some conclusion. If you don't have reasons to support your conclusion, or your reasons to support some conclusion are improbable, then you aren't being rational.
No. The probability is exactly 1. It is impossible to get any other cards except those first twenty you were dealt.
The probability of getting specific cards in those first twenty, is not the same as getting those specific cards in the first twenty.
The probability of getting cards that are completely useless or allow you to win the hand is improbable. You will probably get something in between. The chances of becoming a top athlete or an invalid is improbable. Again somewhere inbetween is more probable. But we are constantly weighing our chances in life.
It looks like I need a different example.
Suppose a friend messages you on Facebook that he just won the Powerball jackpot. The chance of winning the jackpot is 1 in 292,201,338. Do you believe him based on that message? Not sure if we can estimate the probability that your friend is lying, but let's just assume that your friend is not the type to joke like that.
The probability that your friend won the the Powerball jackpot is 1 in 292,201,338. The probability that your friend is lying is likewise is very slim. Either way, you have to choose to believe in something improbable, am I right?
Edit: See this post.
It can be perfectly rational to believe the improbable, you just need rational reasons for doing so.
Whatever the probability of something, there is a fact of the matter of that something. For example, it might be highly improbable to be dealt 5 cards and 4 of them are the same card, Four of a kind but there is a fact of the matter about what cards are going to come up based on the order of the cards in the deck.
You can have rational reasons for believing the improbable. Its highly improbable that life exists on a rock floating through oblivion but none the less that's what happened. In fact, highly improbable things happen all the time, its not irrational to believe those things actually do happen.
You are right that if you dont have reasons to support a conclusion then you arent being rational but I do not think its correct to say that if your reasons support a conclusion thats improbable you are necessary being irrational. Probability is not the same as the fact of the matter of what is.
Sure we can; he is lying or he isn’t lying. No such thing as a partial lie. The probability is exactly .5.
As to whether I’d believe him, I guess I would. I mean....he’d look pretty stupid if he told me he did, but couldn’t prove it. So if I believe no one intentionally wants to make himself look stupid, I’d also have to believe he wasn’t lying. But he could have won and then disappeared, so I wouldn’t know either way anyway.
I suppose you can say that this isn't the relevant probability. The relevant probability is the probability that your friend won the jackpot given that he told you that he won.
Why .5?
You don't have to believe either scenario. All you have to believe is that your friend said it.
You only have to choose to believe one or the other should you need to make a decision or take an action based upon what your friend said.
How do you know how probable the existence of life is? It seems certain that life exists in this universe, not probable.
Probability of 1 is certainty. Only he has the certainty, thus only he knows he won or is lying about winning. If he just tells me he won, I do not have the certainty of knowledge, so I do not have the probability of 1 (he won) or the probability of 1 (he is lying). The very best I can do, without the facts, is split the difference. Splitting 1 in half gives probability for me, of .5.
Why couldn't we just not assign probability in that case? Leave it as an unknown probability.
We could, and most likely would. But there was a question asked, for which an answer is called.
Well it depends on what you mean by probable I guess. Typically people use it to describe something that could happen, not something that did happen.
There is no probability after the fact. For example there is no probability that I wrote that previous paragraph, I DID write that previous paragraph.
When I mentioned the probability of life I was speaking from a normal, probabilistic sense. I wasnt using this “probability after the fact” version, though I can see now I could have been more precise. I should have said “for life to have formed” or “for life to come to exist” instead of “for life to exist”. My mistake, but my points still stand.
Again, just noting that something is improbable is too unspecific. Something, somewhere is always probable or improbable, depending on how you look at it. I am going to do a bit of probability algebra, but before I do I just want to emphasize that the most important thing is not cranking the handle and spitting out formulas, but to be very clear about what it is that you are evaluating - otherwise it's GIGO.
So, what do you know? You know with certainty what you just heard from your friend (let's assume that you are not dreaming or hallucinating):
E = My friend told me that he won the lottery
P(E) = 1
This event could happen in one of two ways: either your friend really won the lottery and he is being truthful, or your friend lost the lottery and he is lying (for simplicity we'll neglect all other possibilities):
W = My friend won the lottery
P(W) = 1/292,201,338 = w << 1 (very unlikely)
T = My friend is telling the truth
P(T) = t ~ 1 (very likely)
What we want to know is the probability of the first of these two disjuncts: P(W ^ T) = ?
1 = P(E) = P(W ^ T) + P(-W ^ -T)
P(W ^ T) = 1 - P(-W ^ -T) = 1 - P(-W)P(-T | -W) = 1 - P(-W)P(-T) = 1 - (1 - w)(1 - t)
Here I made another important assumption: P(-T | -W) ~ P(-T), i.e. my friend's sincerity is unconditional. The converse of that would be that my friend's sincerity can be depended upon only when nothing important is at stake. If we push aside that ugly thought, then we have our result:
I can believe my friend's claim if the probability 1 - (1 - w)(1 - t) is not too low - let's say, if it is greater than 0.5:
1 - (1 - w)(1 - t) > 0.5
(1 - w)(1 - t) < 0.5
Here we have a product of two numbers: (1) the probability of losing a lottery, which is known and is close to 1, and (2) the probability that my friend is lying, which is less certain but is assumed to be close to 0.
So it comes down to how much you trust your friend's truthfulness, but you knew that all along, didn't you? The moral of the story though is that what is rational to believe is what is probable. No exceptions. The trick is to evaluate not just any probability but the appropriate probability.
If you say that the probability is 0.5, then you are saying that you have no more reason to believe one way than the other. And you appear to deduce this just from the fact that there are two disjoint possibilities. Either you are misusing probabilities or you are being unreasonable. (Just try to apply the same logic to the disjoint events of winning or losing the lottery: surely it isn't fifty/fifty?)
I think human reactions are tied to expectations. When a player gets a particular hand s/he isn't surprised because s/he didn't expect that particular hand.
However, imagine the player had a particular hand in mind and was expecting that. In the event that s/he gets what was expected then a reaction of surprise naturally follows because the event is highly improbable.
Similarly the average parent knows how improbable it is for his/her child to become an NBA player and when that happens it is met with pleasant surprise.
In some sense the degree of disbelief or surprise depends on how improbable an event is and how low the expectations were.
When we buy a lottery ticket we're never surprised by the sequence of numbers the ticket you bought has because there was no expectation involved. However, when we win the lottery we're shocked because we never expected to win because of how improbable it is to get the winning combination.
At first glance, getting into the NBA could be just 1 chance out of 1 million or so, over the population of all Americans.
However, it is probably 1 chance out of 100 out of all Americans who train 8 hours per day on getting as far as they could get. It is possibly 1 chance out of 20 for all Americans who are 6 foot 2, train 8 hours per day, and have been doing so since they were 6 years old. It is possible 1 chance out of 5 for all Americans who also have an uncle who managed to get into the NBA.
And so on.
Most people who have been training on getting into the NBA from the age of 6 can quite accurately predict by the age of 16 if they stand a reasonable chance of getting in. That person may already be 70% likely to get in, if he keeps training 8 hours a day with a first-class coach for junior players.
If your intuitions are the same as mine, you might prefer to represent epistemic uncertainty in terms of a set of probability distributions, as opposed in terms of a particular distribution. That way, the controversial use and interpretation of 'prior probabilities' is avoided entirely.
To illustrate, consider trying to catch a fish in a lake with a net. Ideally the net would cover the entire volume of the lake. The net is then 'reeled in' to catch the fish. The initial placement of the net represents the 'prior search space' of the problem, and could be said to represent any 'prior probability distribution' that assigns positive mass over the same area.
I don’t think so. All I was asked was......
Quoting Wheatley
......and the message was merely informing me he won the lottery. I wasn’t asked to determine an answer based on any conditions. Pretty simple really: he said he won, do you believe him. So I do or I don’t, and it makes no difference whatsoever which it is. It can only be one or the other, from which follows the probability of .5 for the answer (my belief), conforming to the fact contained in the question (he won).
Subsequently, if asked why I responded with I do or I don’t believe him, then I would have to condition that answer with other beliefs, but those beliefs would necessarily be based on what I know about the man, not what I know “....based on that message”.
If anything, I’d be unreasonable, indeed irrational, to read stuff into the original query, such that I’m answering questions that weren’t even asked, or, I’m answering under conditions not given in the question that was asked.
But you already live after the fact. I asked you how do you know the probability of life arising in any universe? Doesn't it depend on pre-existing conditions? And if those pre-existing conditions exist, then life is certain to exist, and without them then life is certain to not exist. Probabilities are ideas that stem from our ignorance of what is, was, or will be.
I can see you making that statement before the fact, say 13 billion years ago when galaxies were just beginning to form and before any life existed. You would say that it is improbable for life to form, but that would be based on your ignorance of the future.
Well ya, if I had all the data I wouldnt need probability. I don’t think we disagree significantly here.
No, it doesn't. You are misusing probability.
If you insist, so be it.
Consider an urn containing N balls, each ball being one of K colours. Suppose we express our 'subjective indifference' to the colour c of a ball randomly drawn from the urn by saying that our 'prior belief' is P(c)=1/K for c=1..K. On the surface, this assignment might look objectionable on the grounds that it is the physicality of the urn we are actually interested in, which could contain any proportion of balls of each colour. Yet our supposedly 'subjective prior beliefs', are in fact equivalent to saying that the actual ball colour frequencies inside the urn are undefined in our problem; for notice that the colour frequencies of balls drawn from every possible urn containing N balls of K potential colours, is 1/K.
In other words, in this problem our expression of a 'subjective' and 'definite' prior probability is rational, but it is actually a misnomer, since it is a frequency summary of logical deductions over the set of possible world states defined by our problem,in which the actual world-state is left undefined.
In conclusion, whenever a subjective belief is expressed and explicitly quantified in a problem, it should be understood frequentially in terms of a set of logical deductions over a set of possible worlds.