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The concept of probability is not as simple as you think

By anonymous follow anonymous   2019 Mar 2, 4:08pm 1,672 views   17 comments   watch   nsfw   quote   share    


The gambler, the quantum physicist and the juror all reason about probabilities: the probability of winning, of a radioactive atom decaying, of a defendant’s guilt. But despite their ubiquity, experts dispute just what probabilities are. This leads to disagreements on how to reason about, and with, probabilities – disagreements that our cognitive biases can exacerbate, such as our tendency to ignore evidence that runs counter to a hypothesis we favour. Clarifying the nature of probability, then, can help to improve our reasoning.

Three popular theories analyse probabilities as either frequencies, propensities or degrees of belief. Suppose I tell you that a coin has a 50 per cent probability of landing heads up. These theories, respectively, say that this is:

•The frequency with which that coin lands heads;

•The propensity, or tendency, that the coin’s physical characteristics give it to land heads;

•How confident I am that it lands heads.

But each of these interpretations faces problems. Consider the following case:

Adam flips a fair coin that self-destructs after being tossed four times. Adam’s friends Beth, Charles and Dave are present, but blindfolded. After the fourth flip, Beth says: ‘The probability that the coin landed heads the first time is 50 per cent.’

Adam then tells his friends that the coin landed heads three times out of four. Charles says: ‘The probability that the coin landed heads the first time is 75 per cent.’

Dave, despite having the same information as Charles, says: ‘I disagree. The probability that the coin landed heads the first time is 60 per cent.’

The frequency interpretation struggles with Beth’s assertion. The frequency with which the coin lands heads is three out of four, and it can never be tossed again. Still, it seems that Beth was right: the probability that the coin landed heads the first time is 50 per cent.

Meanwhile, the propensity interpretation falters on Charles’s assertion. Since the coin is fair, it had an equal propensity to land heads or tails. Yet Charles also seems right to say that the probability that the coin landed heads the first time is 75 per cent.

The confidence interpretation makes sense of the first two assertions, holding that they express Beth and Charles’s confidence that the coin landed heads. But consider Dave’s assertion. When Dave says that the probability that the coin landed heads is 60 per cent, he says something false. But if Dave really is 60 per cent confident that the coin landed heads, then on the confidence interpretation, he has said something true – he has truly reported how certain he is.

Some philosophers think that such cases support a pluralistic approach in which there are multiple kinds of probabilities. My own view is that we should adopt a fourth interpretation – a degree-of-support interpretation.

Here, probabilities are understood as relations of evidential support between propositions. ‘The probability of X given Y’ is the degree to which Y supports the truth of X. When we speak of ‘the probability of X’ on its own, this is shorthand for the probability of X conditional on any background information we have. When Beth says that there is a 50 per cent probability that the coin landed heads, she means that this is the probability that it lands heads conditional on the information that it was tossed and some information about its construction (for example, it being symmetrical).

Relative to different information, however, the proposition that the coin landed heads has a different probability. When Charles says that there is a 75 per cent probability that the coin landed heads, he means this is the probability that it landed heads relative to the information that three of four tosses landed heads. Meanwhile, Dave says there is a 60 per cent probability that the coin landed heads, relative to this same information – but since this information in fact supports heads more strongly than 60 per cent, what Dave says is false.

The degree-of-support interpretation incorporates what’s right about each of our first three approaches while correcting their problems. It captures the connection between probabilities and degrees of confidence. It does this not by identifying them – instead, it takes degrees of belief to be rationally constrained by degrees of support. The reason I should be 50 per cent confident that a coin lands heads, if all I know about it is that it is symmetrical, is because this is the degree to which my evidence supports this hypothesis.

Similarly, the degree-of-support interpretation allows the information that the coin landed heads with a 75 per cent frequency to make it 75 per cent probable that it landed heads on any particular toss. It captures the connection between frequencies and probabilities but, unlike the frequency interpretation, it denies that frequencies and probabilities are the same thing. Instead, probabilities sometimes relate claims about frequencies to claims about specific individuals.

Finally, the degree-of-support interpretation analyses the propensity of the coin to land heads as a relation between, on the one hand, propositions about the construction of the coin and, on the other, the proposition that it lands heads. That is, it concerns the degree to which the coin’s construction predicts the coin’s behaviour. More generally, propensities link claims about causes and claims about effects – eg, a description of an atom’s intrinsic characteristics and the hypothesis that it decays.

Because they turn probabilities into different kinds of entities, our four theories offer divergent advice on how to figure out the values of probabilities. The first three interpretations (frequency, propensity and confidence) try to make probabilities things we can observe – through counting, experimentation or introspection. By contrast, degrees of support seem to be what philosophers call ‘abstract entities’ – neither in the world nor in our minds. While we know that a coin is symmetrical by observation, we know that the proposition ‘this coin is symmetrical’ supports the propositions ‘this coin lands heads’ and ‘this coin lands tails’ to equal degrees in the same way we know that ‘this coin lands heads’ entails ‘this coin lands heads or tails’: by thinking.

But a skeptic might point out that coin tosses are easy. Suppose we’re on a jury. How are we supposed to figure out the probability that the defendant committed the murder, so as to see whether there can be reasonable doubt about his guilt?

Answer: think more. First, ask: what is our evidence? What we want to figure out is how strongly this evidence supports the hypothesis that the defendant is guilty. Perhaps our salient evidence is that the defendant’s fingerprints are on the gun used to kill the victim.

Then, ask: can we use the mathematical rules of probability to break down the probability of our hypothesis in light of the evidence into more tractable probabilities? Here we are concerned with the probability of a cause (the defendant committing the murder) given an effect (his fingerprints being on the murder weapon). Bayes’s theorem lets us calculate this as a function of three further probabilities: the prior probability of the cause, the probability of the effect given this cause, and the probability of the effect without this cause.

Since this is all relative to any background information we have, the first probability (of the cause) is informed by what we know about the defendant’s motives, means and opportunity. We can get a handle on the third probability (of the effect without the cause) by breaking down the possibility that the defendant is innocent into other possible causes of the victim’s death, and asking how probable each is, and how probable they make it that the defendant’s fingerprints would be on the gun. We will eventually reach probabilities that we cannot break down any further. At this point, we might search for general principles to guide our assignments of probabilities, or we might rely on intuitive judgments, as we do in the coin cases.

When we are reasoning about criminals rather than coins, this process is unlikely to lead to convergence on precise probabilities. But there’s no alternative. We can’t resolve disagreements about how much the information we possess supports a hypothesis just by gathering more information. Instead, we can make progress only by way of philosophical reflection on the space of possibilities, the information we have, and how strongly it supports some possibilities over others.

https://aeon.co/ideas/the-concept-of-probability-is-not-as-simple-as-you-think

#Probability #Logic #PhilosophyOfScience

1   anonymity   ignore (0)   2019 Mar 2, 7:56pm     ↓ dislike (1)   quote   flag      

This article is nothing but pseudo intellectual garbage.

I have an advaneed degree in statistics and can tell you this is just nonsense jibberish intentionally written to confuse and sound 'intelligent'. I can assure you that the concept of probability is far more concrete than what this author believes. Clearly he needs to spend more time learning about probability; the only person confused about this topic is him.
2   ad   ignore (0)   2019 Mar 2, 8:50pm     ↓ dislike (0)   quote   flag      

If the coin is fair, then there is a 50% chance it will land on heads, and 50% chance it will land on tails. There is nothing more complicated to understand about the probability event involving a coin, or let alone the throwing of 2 fair dice.
3   mostly reader   ignore (0)   2019 Mar 2, 9:27pm     ↓ dislike (0)   quote   flag      

anonymity says
This article is nothing but pseudo intellectual garbage.

I have an advaneed degree in statistics and can tell you this is just nonsense jibberish intentionally written to confuse and sound 'intelligent'. I can assure you that the concept of probability is far more concrete than what this author believes. Clearly he needs to spend more time learning about probability; the only person confused about this topic is him.
+1
The author is deeply confused about the subject, to the point that (s)he doesn't understand the difference between basic concepts. Such as probability and likelihood.
In nutshell, the difference is this.
Probability: pertinent to coin flip scenario. It can be studied because it deals with events that can be repeatably (and infinitely) reproduced in controlled environment.
Likelihood: in all likelihood, the OP doesn't understand what (s)he is posting. My money would be on it. However, this bet would be unscientific because probability theory wouldn't attempt to quantify this crap.
4   Ceffer   ignore (6)   2019 Mar 2, 10:45pm     ↓ dislike (1)   quote   flag      

"The page you were looking for could not be found (404)." Now, what's the likelihood of that?

Article's intended purpose is to murk, not enlighten, by making probability a matter of subjective judgement, and therefore, more fallible than it actually is. This appears to be for the purpose of confusing a shallow and gullible audience.
5   anonymous   ignore (null)   2019 Mar 3, 2:00am     ↓ dislike (0)   quote   flag      

Ceffer says
"The page you were looking for could not be found (404)." Now, what's the likelihood of that?


Pretty high, really fucking high now that you mention it considering the formatting changes Patrick did to highlight the word you or your which affects anything you search for with those words in the title. That is why the link does not work but the article is sitting there waiting for the ambitious.

Never crossed someone's mind to go to https://aeon.co/ and look for the article - my bad - that would require initiative instead of being spoon fed
6   anonymous   ignore (null)   2019 Mar 3, 2:05am     ↓ dislike (0)   quote   flag      

anonymity says
I have an advaneed degree in statistics and can tell you this is just nonsense jibberish intentionally written to confuse and sound 'intelligent'. I can assure you that the concept of probability is far more concrete than what this author believes


Go for it - I am eager to read this dissertation. Let's see that concrete thesis. (I'll check back often to see the progress)

Assure me mighty one - please for the love of all that it is good in the world - assure me.

No one caught on (including "anonymity" who just coincidently joined the forum today) that this article was less to do with probability and mostly about inherent bias than anything else especially when weighing the odds of someone being guilty in a criminal case etc.

Here's your sign....
7   marcus   ignore (11)   2019 Mar 3, 11:36am     ↓ dislike (0)   quote   flag      

anonymity says
This article is nothing but pseudo intellectual garbage


Garbage might be a little extreme. It is definitely all over the map, without a very clear point he's trying to make - and perhaps too much philosophy lingo (maybe that's the audience). Ultimately I guess it's supposed to be about applying Math or not applying Math to questions of innocence or guilt in court cases ? Conditional probability (bayes theorem: i.e. probability of A given B) ) can sometimes be applied to real questions to give answers that are on the surface counter intuitive, and definitely has courtroom applications, either in support or in rejection of a prosecutors case. But they would have to do a little mini course for the jury to bring them up to where they might understand.
8   marcus   ignore (11)   2019 Mar 3, 11:45am     ↓ dislike (0)   quote   flag      

Kakistocracy says
The probability that the coin landed heads the first time is 75 per cent


This does get slightly in to a mind bending area. You have something with a .5 probability that happened 3 out of 4 times. So to ask after the fact what is the chance that the first one was heads, would obviously be .75.

This reminds me of the "Sleeping Beauty Problem" which is fun, and clearly the concept the author is alluding to (not very well - or perhaps there is an assumption the reader knows this since he touches on it so briefly).


Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep.

Once or twice, during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

If the coin comes up heads, Beauty will be awakened and interviewed on Monday only.
If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.
In either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Beauty is asked: "What is your credence now for the proposition that the coin landed heads?"


That is, she is asked what is the probability that the coin landed heads ?

https://en.wikipedia.org/wiki/Sleeping_Beauty_problem
9   marcus   ignore (11)   2019 Mar 3, 12:07pm     ↓ dislike (0)   quote   flag      

Is it more likely from SBs pov that it's Monday rather than Tuesday ? Shouldn't it be ? And if so, does that make it less likely to be tails,rather than heads ? Or is it the other way around ?
10   mostly reader   ignore (0)   2019 Mar 3, 2:38pm     ↓ dislike (0)   quote   flag      

marcus says
Is it more likely from SBs pov that it's Monday rather than Tuesday ?
Yes
And if so, does that make it less likely to be tails,rather than heads ?
No
11   Shaman   ignore (2)   2019 Mar 3, 2:53pm     ↓ dislike (0)   quote   flag      

His idea would work better with predictive probabilities, where the observer doesn’t know the actual probability (unlike a simple coin flip) but must infer the probability of a certain outcome based on limited observations. Say, the probability that a Chinese restaurant chosen at random will have good food. We can’t try them all, but we can try a few and extrapolate a probability from our results.
Say we try five and three of them had good food. That’s 60% chance. However, one should account for the possibility that our sample size was too small, and the next three Chinese places will be awesome. So we weight the “discovery” side of the equation a bit, say 10% to the chance that the next one we try will have bomb lo mein.
That’s how we arrive at a working probability of 70% that the next random Chinese place we visit will have good food.

We do these sorts of calculations all the time, and the more information we know about the subject, the better we can predict our odds of success, and the better we can plot our path through life!
12   ad   ignore (0)   2019 Mar 3, 5:00pm     ↓ dislike (0)   quote   flag      

Yes, this deals with stochastic processes like modeling and forecasting.

Wall Street employs it to make financial decisions.
13   FuckTheMainstreamMedia   ignore (7)   2019 Mar 3, 5:04pm     ↓ dislike (0)   quote   flag      

I THINK what the author of OP is trying to do (and thereby show his ignorance) is to claim that probabilities and odds are affected by unknown factors.

As the first few follow ups posted its an utter pile of cow manure.

Look at it this way....

In Texas Hold Em Poker there will be a 1 in 8 chance that if you hold a pair in your hand, a matching 3rd card will appear on a 3 card flop. That is, you know two of the cards in the 52 card deck. So there are two remaining which means you calculate as 2/50 + 2/49 + 2/48.

I once argued with a guy who was adamant that this cannot be possibly true since one or both of the remaining cards could be in an opponents hand.

I tried explaining that while that is true, you can't possibly know one way or the other. Therefore you must make the calculation as though all the cards remain in the deck.

The guy could not wrap his head around it. Incidently that was the day I finally woke up and stopped giving lessons at the table.
14   HeadSet   ignore (2)   2019 Mar 3, 5:36pm     ↓ dislike (0)   quote   flag      

I finally woke up and stopped giving lessons at the table.

The only lesson you should give at the Poker table is to watch you demonstrate how to scoop the pot....
15   Reality   ignore (8)   2019 Mar 3, 5:52pm     ↓ dislike (0)   quote   flag      

The article's author was very confused, and obviously didn't understand the concept of conditional probability. Here are a couple thought experiments:

1. Suppose, it is revealed that all 4 coin flips yielded heads. What's the probability of the first flip was head? Obviously 100%! Therefore, when it was revealed that 3 out 4 outcomes were heads, the 1st of the 4 flips had 75% probability of being head.

2. When you buy a lottery ticket that has 50million different combinations, what's your winning probability before drawing? 1/50million; what about after drawing and revealing the winning number is different from yours? Obviously 0%.

The author of the article was confusing himself quite unnecessarily.

As for his logical extension to murder trials, here's the real kicker: Do you the probability of murder cases being solved in cities like Detroit and Chicago? 15%! In other words, a murderer has 85% chance of getting away with murder. Seems to be a very strong case for recommending everyone owning their own guns for self-defense . . . as the law enforcement there is quite toothless as far as providing protection from murder is concerned . . . how a jury should examining murder evidence in those places is 85% irrelevant, as the overwhelming majority of murder cases don't arrive at that stage. Only a higher statistical probability of would-be victim having the means to shoot back would provide a strong deterrence in those places.
16   Goran_K   ignore (3)   2019 Mar 3, 5:58pm     ↓ dislike (2)   quote   flag      

The author sounds like a liberal arts grad who is still paying off his school loan at 42 years old.
17   anonymous   ignore (null)   2019 Mar 3, 7:14pm     ↓ dislike (0)   quote   flag      

Goran_K says
The author sounds like a liberal arts grad who is still paying off his school loan at 42 years old.


No - this is not directed at anyone in particular - just personal experience dealing with MBAs. Most MBAs would be much better served in life by becoming or adhering to MBWA

What does MBA stand for?

MBA stands for Management by Assholes

https://www.acronymattic.com/Management-by-Assholes-(MBA).html (Acronym Finder has 90 verified definitions for MBA)

MBA ! - MBA! - MBA! must have - must get - must be


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