weird probability

[ebon dew]

i already know the answer is 1/11, but i don't get why.
i figured, if you know one dice is gonna be 3, then for both to be 3, then the other dice has a 1/6 of landing on 3, so it's 1/6

[lament meadow]

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[ebon dew]

this gyu drew a table tho

]

[minor oxide]

That’s how it should work.

Since it would either be a 3 on the first die or a 3 on the second.

[slate roost]

Are you familiar with Bayes's theorem?

[keen meadow]

Conditional probability.
P(both 3 | at least one 3) = P(both 3 ⋂ at least one 3)/P(at least one 3) = P(both 3)/P(at least one 3)

[keen meadow]

Oh, yeah, that's a nice approach, too.

[ebon dew]

nah i've been meaning to learn it tho

[ebon dew]

they don't teach this notation in the uk until 17

[keen meadow]

What notation?

[ebon dew]

tbh idek if they teach it then

union and intersection and the bar

[minor oxide]

What kind of combinations would result from this?

[keen meadow]

But it's a conditional probability problem.

How would you write it, then?

[slate roost]

I mean, I'm a fan of Boolean algebra notation.

[ebon dew]

i think i know

wait

[minor oxide]

Also, you can think about it like this: the permutations “reflect”, i.e., (1,3) is different from (3,1). Thus we get 2*6-1=11 possible probabilities, and only one of these is (3,3), since reflexivity in this case would not work on it.

[slate roost]

There's two fundamental rules of probability theory you need to know to basically be able to do anything possible. They are: P(AB) = P(A|B) * P(B) = P(B|A) * P(A) P(A + B) = P(A) + P(B) - P(AB)

@ebon dew You still with us?

[ebon dew]

at gcse in england they only teach probabiltiy trees so i just tried to represnet it using that and failed

[ebon dew]

idk wht the `¦ means

[slate roost]

It means "given".

[minor oxide]

Tables suffice here. They’re useful regardless, but it’s good to know the notation.

[slate roost]

P(A|B) means "the probability of A given B" or "the probability of A if B is true."

[slate roost]

@ebon dew

[keen meadow]

Oh, wait, so this is supposed to be solved just using the definition of probability?

Well, that kinda sucks...

[minor oxide]

That’s what it looks like.

[slate roost]

I mean, there's no indication of how it's "supposed" to be solved.

And we're not getting a lot of feedback from the helpee about what exactly confused them.

[keen meadow]

Yeah, true, but seeing as what Max said about notation and stuff, that's what I thought.

[minor oxide]

Yeah, there isn’t necessarily an off limits method. It is only a question with no confined boundaries, i.e., in a textbook.

[keen meadow]

I mean, it's possible to do it like that, of course, but that's so inefficient...

What if the dice had 20 sides?

[ebon dew]

sorry

it's just the new vocab and stuff, i understand now why my initial reasoning was wrong, but now it's about coming up with an expression for the probabilty

[slate roost]

I'm not sure what you mean by that.

[ebon dew]

Not a single word was understood

I’ll rephrade

Rephrase

I see why I cannot just assume the first dice is the one that rolls a 3, so i get that my idea of 1*(1/6) was wrong

But idk how to form an expression for the probability now

[minor oxide]

If you don’t know the notation, create a table.

[slate roost]

I don't know what you mean by that.

[minor oxide]

It follows that you have, from the first roll of a 3, the ordered pairs (3,1),(3,2)...(3,6).

[ebon dew]

Ye I mean I’m preparing for an Olympiad style thing tomorrow so I’ll know now to just make a table if something like this occurs but now it’s just interest

[minor oxide]

Where the (x,y)->(first roll, second roll).

[minor oxide]

Create the listing for the 2nd rolls, and make sure you don’t overcount.

[ebon dew]

You don’t know that the first roll is 3 though?

[minor oxide]

This is casework.

[slate roost]

Look, I think I can help correct whatever your mistake was in your probability tree.

[ebon dew]

Ohh I think I see

Actually idk

[minor oxide]

Why?

[slate roost]

Look, you need to tell us how you're thinking about the problem in order for us to help you.

You need to work with us.

[ebon dew]

ok

righty then

ill start over because i didnthave much progress anyway

[minor oxide]

Ok.

[ebon dew]

this was my original idea, but i understand how it's wrong n ow

[minor oxide]

Why is it wrong?

[ebon dew]

cause it could be the second dice that is 3

[minor oxide]

👍

Now, what kind of a novel approach should you take?

[ebon dew]

this is wht i meant by probability tree

[minor oxide]

Hmm.

[slate roost]

Okay, so now which branches contain at least one 3?

[ebon dew]

top 3

[slate roost]

So we only want to pay attention to those branches.

[ebon dew]

let me label them so i cna reference them easier

[keen meadow]

Oh? What's this?

[ebon dew]

its intended for like 14/16 year olds

[keen meadow]

Hm... I don't think I've ever seen this method.

[slate roost]

Look, probability theory isn't that bad.

[ebon dew]

idk this might be an unconventional problem for it

[slate roost]

It all comes down to these two rules.

[ebon dew]

ik but i hve had no formal education on it aside from "probability trees", i just self study everything, and usually it does not include probabilty, and more so calc and lin algebra

[ebon dew]

yea ik but this olympiad thing is done by 13 year olds and i dont think they learn this

[ebon dew]

here, can i make branch 3 a 6/6 chance because we know we always get a 3

[slate roost]

They do if they self-study it.

[ebon dew]

the test is based on things they would learn tschool,

[slate roost]

...these rules are probability theory. Like, they're exactly the same thing that you're doing in your little probability trees, but, like, algebraically, and so way more convenient.

[ebon dew]

i dont think the averge person sitting this test would understnd any of those symbols, but i know

[keen meadow]

That's a bid odd, though. The notation isn't difficult, why not have it in school? You are going to use it later in university, anyway, so then there wouldn't be a need to relearn it.

[ebon dew]

similra to how we were taught the power rule without knowing that it follows first principles or what any of it actually means on the intuitive level

[minor oxide]

Odd.

[ebon dew]

uk school has basiaclly like 11-16 gcses, 16-18 a levels, then uni

at a levels, i think they teach you it

but im 15 so i have to learn it all myself

[minor oxide]

But, like, it’s universal notation, and also is pretty easy to interpret.

[keen meadow]

Sorry, I don't really know what those mean.

[ebon dew]

tbf we learn union and the upside down union in sets

[ebon dew]

dw

[minor oxide]

$\cap$ and $\cup$ in set theory?

[halcyon hollow]

Temu Mod

[ebon dew]

as in, set1 = [1,2,3,4,5], set2=[1,3], nd they woiuld make u draw a venn diagfram and find the union

[ebon dew]

yea

[minor oxide]

And you haven’t learned probability notation yet?

[ebon dew]

i disagree with it

[keen meadow]

Oh, that's even more odd, then... Usually you learn them at the same time, roughly.

[ebon dew]

oy

[minor oxide]

Yeah.

[ebon dew]

it's hardly set theory i mean

[minor oxide]

I learned mine last year.

[ebon dew]

idk set theory but i assume its more than

"put these numbers in this circle, and the other numbers in the other circle"

[keen meadow]

It still is, though. It's operations on sets, so, set theory.

[minor oxide]

It’s some part of it.

[slate roost]

You disagree with it?

[ebon dew]

with the order we learn things

[slate roost]

Look.

Here's Boolean algebra.

Or, wait, before that.

You do understand, like, Boolean logic, right?

[ebon dew]

ytea i can code

[slate roost]

Okay, so.

In Boolean algebra, the Boolean product corresponds to "and", and the Boolean sum corresponds to "or".

[keen meadow]

Oh, actually, sorry to divert the conversation, but I'm just really curious.
If you want to write the probability of thins A and B happening at the same time, would you write it as P(A ⋂ B) or as P(A ∧ B)? As in, do you view events more like sets or more like statements about things?

Just curious.

I prefer the set approach, by the way.

[random kraken]

They teach this in the o lvl syallbus too

[main sage]

@ebon dew

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