Hacker News Reader

Comment by tqi | Hacker News Reader

I think this is less about Bayesian thinking and more about misinterpreting the question, as another poster mentions: https://twitter.com/farrwill/status/1751788706355392639

ie i think many of the More Likely to be Green people are doing the math of if you pull from an urn with n/100 odds of getting red, your second pull will have odds (n-1)/99, which is less than n/100 for all n except n=100. Which is obviously a different question than should you bet on Green or Red.

Which is obviously a different question than "would you bet on the next ball being red or green"

View on HN

Why would n be "uniformly randomly chosen" but then suddenly be known? I would mark this mis interpretation simply as wrong if I was a teacher.

That's definitely fair, and I assuming in a classroom setting you'd be very consistent and precise about what that means. But this is a twitter poll, and I doubt people are that consistent. Ultimately I'm just pushing back on the conclusion that this is because "Bayesian thinking is really foreign to people."

I think if the question had instead been "would you bet on the urn having started out with more red balls, green balls, or equal number of balls" the distribution would have been very different.

That question is quite a bit of handholding. A closer analogue would be “would you bet the next ball is red / green / they’re equally likely?”.

Personally I’d word it as “The next ball is… more likely to be red than green / more likely to be green than red / equally likely to be red or green / I don’t know.”

But isn't that exactly how it is worded? Just showing it a bit?

Not quite. People are arguing that “more likely to be red” can be interpreted as “more likely to be red than it was last time”. I guess you could do that with this wording, but that seems like quite a stretch.

i personally would throw out the question if i was a teacher. i still feel it's ambiguous

Another way of making your point: it would also make the information about the distribution of n irrelevant, so it's a bad interpretation of the question.

So? There's plenty of problem statements with irrelevant data. Children are also getting taught to determine which is relevant in class.

But also, it's not irrelevant because for two cases the probability stays equal. Calculating the exact chance is still an interesting problem and it could have been dumbed down for twitter only.

In those cases presumably there's not a more interesting question that's consistent with the question and that uses that information. I'd start with the assumption that the tweet isn't a dumb trick question.

Because including some pieces of irrelevant information as red herrings in probability problems is totally unheard of?

You have an urn of 50 red and 50 green balls. Roll a die and add a red ball for every pip. What is the probability of the next ball you take being red?

I say both can be a question in Statistics I where you calculate probabilities based on combinations.

So I'll say it depends on the level of education for how the question gets interpreted and saying those who never took Bayesian reasoning and consider n unknown are wrong gets elitist.

Not known, but certainly defined. It has to not be zero at least, otherwise you couldn't take out a red ball in the first place. Or at least that's how I understand this problem. You don't know what number n is, so you need statistics or simulations to grasp the probability of the color of the next ball.

I'd argue that it's not so much a difference of opinion than it is just a reasoning error given the question as stated. That's sort of the whole point of this post-- this is a case where "doing the math" in the expected way gives the wrong answer, because the state of the system is cast in stone (in terms of the ratio of red/green balls in the urn) when you first start out, so it's all about leveraging the information you get from your first choice. I think this sort of reasoning is actually surprisingly useful and widely applicable in everyday life, like in investing or diagnosing the likely cause of a software bug from incomplete information.

It's definitely a poorly worded question. "More likely" is ill-defined. More likely than what? Than the last draw? Than drawing the other color just on this pick?

More likely than the other option, given the information. Getting an accurate grasp on the information is the whole point of the question.

Why is that the question? Why not "more likely than the last draw" ?

How could the next draw be more likely to be red than the draw already known to be red?

You are basically asking "how could it possibly be the wrong answer?".

Yes, under that definition of "more likely", it cannot be more likely. It can be equally likely or less likely. And then weighted it would be less likely. And that would be the answer. Which is one of the choices.

No, I'm saying "Your interpretation of the sentence is obviously wrong". With that interpretation, parts of the question are nonsense. The more straightforward interpretation does not have that problem.

1 - I said it's poorly worded, I didn't interpret it that way. I could see how people would misinterpret it (as pointed out by the parent comment) and people did misinterpret it that way. Look at the other posts here. If someone posts a question in english and a bunch of english speakers misinterpret the question, it's poorly worded by definition.

2 - Exactly what parts of the question are nonsense under that interpretation?

> How could the next draw be more likely to be red than the draw already known to be red?

Under this interpretation, the second ball is more likely to be green.

I had the same question but the interpretation they're considering is whether the next ball you draw will be more likely to be red than the red ball you drew was likely to be red before you drew it but after the total number of red balls was already determined.

Like: you have a box with a whole bunch of red and green balls. Would removing a red ball increase or decrease the likelihood of drawing a red ball from the box in the future?

I wouldn't buy that as a reasonable misinterpretation of the question in most cases, but I don't know what standard of analysis holds for Twitter polls.

I think it's a matter of knowing that the phrase "where n is chosen uniformly at random" means something very different in a probability question than "where n is an arbitrary value".

If, hypothetically, the majority of words communicated from one homo sapiens to any other homo sapiens (majority weighted by the number of homo sapiens who will hear, read, or otherwise receive the word over the course of the word's existence) were designed to be precise and unambiguous to the extent that the mean number of homo sapiens who receive the word correctly interpret the exact meaning of the word intended by the entity that chose the word is greater than 99.99%, then I would estimate a greater than 90% probability that the cumulative unique information (in the information-theoretic sense) contained in communications correctly interpreted by the receiving homo sapiens on any given day would be less than the cumulative unique information that is currently being correctly interpreted today (January 30th 2024 in the Gregorian calendar), and furthermore the reduction in information thus communicated would have a negative effect on progress towards the majority of cumulative goals implicitly created by all living homo sapiens.

Or I could just say "I think most people interpreted the tweet correctly, and it would be annoying if it was overly precise"

'What is the most likely color for the next ball selected?'.

Try this book out, it's a fun read on how minor changes in words can make big differences in readability: https://www.amazon.com/Sense-Style-Thinking-Persons-Writing-...

Yes, it's a bit of a pet peeve of mine to use the word "more" while leaving vague the point of comparison. Especially in something like a twitter poll -- twitter isn't exactly renowned as a place where people think closely about things before clicking.

How is "more likely"I'll defined? They are two options red or green. Are you more likely to draw red or are you more likely to draw green.. how is it poorly worded?

Because it could be more likely red than green, or more likely red than it was likely that the first choice was red (which is where some people got confused, they thought the question was asking about the relative chances of red on the first choice v second choice), more likely green than the first one was likely to be green(same basic problem).

My first interpretation of "more likely" was also "more likely than the previous ball". It makes more sense in the context of the previous sentence.

For example:

> The first time you shoot a basketball, you will almost certainly miss. The second time, you are more likely to make it.

Does the second sentence mean "more likely to make it than to miss"? No.

Great example.

You don't need to think in terms of information to get the right answer, you just need to be careful of the setup.

You can easily apply frequentist probability reasoning and still get the right answer. You have to start from P(n > 50 | first pull is a red ball), and compute P(second ball red | first ball red) based on that.

Basically the mistake is computing the frequency over 100 balls, instead of computing it over the 100 possible values of n multiplied by the 100 balls.

Yes, you make a good point.

Reply on news.ycombinator.com