Nono
#help-26
smoky owl
wait that's true, then we get integral of e^x (1+x)/ -(1+X)dx, we cancel out 1+x, so integral of -exdx, which would just be -e^x +C?
topaz sinewBOT
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safe knot
i need help
topaz sinewBOT
i need help
safe knot
what do i do
torpid sparrow
Asking your question is a good begining
rose sierra
🤡
lone nest
Very well, please open an available help channel when you're ready to ask a question.
.close
topaz sinewBOT
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lone nest
Are you ready to ask a question now?
rose sierra
No selfie allowed
bro thinks he's clever
wintry current
What is the question?? Please don’t open channels if you aren’t going to answer a question ❤️
topaz sinewBOT
@safe knot Has your question been resolved?
woven sigil
topaz sinewBOT
woven sigil
,rotate
thorny flameBOT
woven sigil
What am i doing wrong ?
restive inlet
the cancellation of sin(x) - cos(x) like that isn't valid
woven sigil
Aw
cedar wagon
What am i doing wrong ?
Factor the denominator as
a^2 - b^2 = (a+b)(a-b)
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toxic aspen
(A \ B) \ C = (A \ C) \ B
Is this statement true?
topaz sinewBOT
(A \ B) \ C = (A \ C) \ B
Is this statement true?
ivory sorrel
try drawing a venn diagram
toxic aspen
Is there mathematical operations i can do to figure out if this is true
Yeah i did, and i came to the conclusion that they are indeed equivalent
rigid ivy
(A \ B) \ C = (A \ C) \ B
Is this statement true?
Sets, or numbers?
rigid ivy
No it does not
toxic aspen
So the only solution would be by showing with a venn diagram?
rigid ivy
But I suppose you could define $A\setminus B=A\cap \overline{B}$
thorny flameBOT
SWR
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frigid salmon
any help with this DFA?
frigid salmon
What is the question?
what does it represent?
not really sure since there's so many
what does it accept*, I should say
urban grove
ah
Can you list some example strings that it accepts?
Maybe we could figure out a general pattern together
frigid salmon
Can you list some example strings that it accepts?
00, 000, 0000
urban grove
right
frigid salmon
011, 0111, 011111
frigid salmon
0, 01, 010, 0101, 01010, 001
frigid salmon
the last part is more stabilized for the accepted?
urban grove
yup!
urban grove
that's almost right
but you also have to account for if the string is not at least two characters long
so it could be empty, 0, or 1 as well
frigid salmon
in that case, it looks like none will accept
so then
size >= 2, end no 10 or 01?
frigid salmon
size >= 2 and must have 00 and 11 at the end
urban grove
perfect! and actually size ≥ 2 is redundant there, so you can just say "must have 00 and 11 at the end"
frigid salmon
oh yeah smart since thats implicit ig
urban grove
so now you can think about what each of the states represent, e.g. you're in state q3 when the previous two characters were both 0
frigid salmon
ok lets see
urban grove
and once you've proven that, it's simple to prove that the DFA accepts exactly the strings that you identified, the ones with 00 or 11 at the end
frigid salmon
so q0 looks like emptystring
q1 looks like 0?
q2 looks like 1
q3 looks like 00
urban grove
yup
frigid salmon
q4 looks like 11
urban grove
in particular, q1 is 0 except if there's another 0 before it
so you end up at q1 with the strings 10, 010, 000010, 0, etc etc
frigid salmon
oh shoot i forgot theres like a loop between q1 and q2
so its not just 0 / 1
for q1 and q2
urban grove
well maybe we can reason through it
when would you take one of those arrows?
that you were confused about
what's an example string that uses them
frigid salmon
like ik how to get an example to get to q1
sm like 0
or 010
or 01010...
but then we got that other edge too
so can also be like 0010
urban grove
yeah, so q1 also accounts for strings that end with 10
bc you wouldn't want that to go to q3, which tracks strings ending in 00
frigid salmon
i think q1 is either 0 or strings that end with 10 ?
q2 is either 1 or strings that end with 01?
urban grove
yup
urban grove
well, what are the different operations in regex?
frigid salmon
i think we use the star one
but idk how to do it if there are 2 possibilities i think
since we either accept 00 or 11
urban grove
how would you write it if you just wanted strings that ended in 00
frigid salmon
how would you write it if you just wanted strings that ended in 00
so definitely ...00
where the ... is like
any combo of 0's and 1's?
urban grove
yeah, IDK the notation that your course uses for regex, but I would probably write that as .*
so, .*00
urban grove
yeah
so the way you usually write that in regex is with a vertical bar |
meaning or
so it'd be like .*(00|11)
check your course notes for the specific notation you're supposed to use though
topaz sinewBOT
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frigid salmon
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neon iron
Hi
neon iron
I have one question
We know that if we have two Lebesgue measurable sets on IR, their union and intersection is also Lebesgue measurable
So my question is if the same happens in IR³
If it's not possible, is there a characterization for when a union of 2 Lebesgue measurable sets in IR³ is measurable?
jade thunder
well the measure on R^3 is a measure so it surely must satisfy the union axiom
worthy storm
how are you defining lebesgue measurable sets?
neon iron
how are you defining lebesgue measurable sets?
I'm defining them as the following
torpid sparrow
No need for additional
neon iron
A set $A \subset \mathbb{R}^n$ is Lebesgue measurable if its outer Lebesgue measure holds the following:
$\forall E\subset \mathbb{R}^n: \mu^(E) = \mu^(E \cap A) + \mu^*(E\cap A^c)$
thorny flameBOT
математик 🇨🇿
worthy storm
the caratheodory condition
one can show with a fair amount of work that the collection of sets that satisfy this condition form a sigma algebra
regardless of whether they live in R^2 or R^3 or whatever
neon iron
one can show with a fair amount of work that the collection of sets that satisfy...
This would solve the question
neon iron
one can show with a fair amount of work that the collection of sets that satisfy...
So now what I need to show is this
worthy storm
showing that the union of two measurable sets is measurable is already a bit of a chore
neon iron
regardless of whether they live in R^2 or R^3 or whatever
Well, I asked this because I'm writing a book in math and I was constructing the concept of volume in IR³
worthy storm
eg. the proof of just that part is about half a page in my measure theory book
neon iron
eg. the proof of just that part is about half a page in my measure theory book
It doesn't matter as long as it can be written based on preliminaries, so it's okay
worthy storm
the full proof that the measurable sets form a sigma algebra is roughly 2 pages
it involves several very clever manipulations which always impress me when i read the proof
neon iron
Well
I thought of something
As I only need to prove that the union of 2 measurable Lebesgue sets is measurable and the same for intersection, there's no need in proving that the set forms a sigma-algebra. We can just prove that it is an algebra of sets
Which is shorter
Okay, I got the help I wanted, thanks
.close
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ocean timber
alrigheady closed man
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lucid junco
I don’t understand what they did to get to the next step in the highlighted bit
lucid junco
Positive
lucid junco
But -1 square is 1
coarse tusk
sin cannot be less then 0
Why can’t sin be less than 0?
coarse tusk
It can be
brave oyster
not equal to 0 i guess
coarse tusk
they just wrote it there
They didn’t
coarse tusk
They said $-1\le\sin\theta\le 1,\sin\theta\ne 0$
thorny flameBOT
kheerii
lucid junco
(-1)^2 = 1 so why is the left restriction changed to 0
coarse tusk
$0<\sin^2\theta\le 1$
thorny flameBOT
kheerii
brave oyster
(-1)^2 = 1 so why is the left restriction changed to 0
after square what is new mininum
coarse tusk
(-1)^2 = 1 so why is the left restriction changed to 0
You cannot square inequalities
Directly
lucid junco
after square what is new mininum
1
brave oyster
wait wait
lucid junco
I don’t know
brave oyster
ok in that range
coarse tusk
Consider two cases
coarse tusk
$-1\le\sin\theta<0$ first
thorny flameBOT
kheerii
brave oyster
yeah lets do that
coarse tusk
Do you think you can just square all sides of this inequality?
lucid junco
I don’t see why not
torpid sparrow
You might understand better with an easier example
coarse tusk
Even powers become weird when you start dealing with negative numbers
If we consider the other case, $0<\sin\theta\le 1$ you absolutely can square all sides
thorny flameBOT
kheerii
coarse tusk
Because the whole interval is positive
torpid sparrow
0 > -1;
0^2 > (-1)^2 (as you can see this is false)
coarse tusk
lucid junco
Oh okay
coarse tusk
$a>b>0\implies a^2>b^2$
thorny flameBOT
kheerii
torpid sparrow
Always 🥔
coarse tusk
$a<b<0\implies a^2>b^2$
thorny flameBOT
kheerii
coarse tusk
If $a<0<b$ we can’t compare $a^2$ and $b^2$ at all
thorny flameBOT
kheerii
lucid junco
So what happens when we square -1<sin<1
coarse tusk
So what happens when we square -1<sin<1
It’s easier to see if you split it into two cases
-1<=sin theta < 0 and 0< sin theta <=1
lucid junco
So what do we do then?
coarse tusk
Now you can square all sides of either expression
Since each individual interval is either fully positive or fully negative
However, when the whole interval is negative the inequality signs flip
lucid junco
But -1<sin<0 involves negative
coarse tusk
The basic idea here is that you wish to compare |a| and |b|
coarse tusk
But -1<sin<0 involves negative
Yeah, which means when you square both sides the inequality flips
lucid junco
For the whole thing?
coarse tusk
Yeah
topaz sinewBOT
@lucid junco Has your question been resolved?
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neon iron
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neon iron
Here I’m learning how to simplify square roots
This YouTube video says choose the highest perfect square that multiplies into the number under the square root
How do I know 16 times 3 equals 48 without a calculator since I don’t know the multiples of 16
So I would have to divide 48 by every perfect square lesser than 48 to find out
And I think this is a poor way of simplifying it
Is there a better way to find out like 16 times 3 is 48
Or should I start memorizing multiplication tables of all perfect squares up to like 100?
I want to gain intuition on simplifying larger square roots without dividing the number by ever perfect square in my calculator to find the answer
snow nimbus
Is there a better way to find out like 16 times 3 is 48
I mean there isn't a method that always works. Especially for really large squares you won't see the prime factors immediately. In case of 48 one should know that 16 divides it though.
But if you don't I would look at the number and see: Oh, it is divisible by 4. Then I'd do 48 = 12 * 4. Again one immediately sees that 12 is divisible by 4.
Don't know if that helps you at all?
neon iron
Yes thank you
How do you know 48 is divisible by 16
I only know multiplication up to 12
Is it because you know the multiples of 16 or something else
snow nimbus
How do you know 48 is divisible by 16
Idc, probably because I calculated 16*3 before
snow nimbus
I only know multiplication up to 12
I think you don't need to know every multiple of every square by heart. If you don't just check starting by the smallest square and factorize
neon iron
Is the best method just to start dividing by perfect squares
Until you find one that works
neon iron
I think you don't need to know every multiple of every square by heart. If you d...
Okay
snow nimbus
Until you find one that works
I would say it is factorizing.
topaz sinewBOT
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long fog
topaz sinewBOT
whole geode
So is a train's route a single diagonal? Or is it any 23-gram? Or something else?
A single train to each route?
long stirrup
looks simple
a diagonal
so if he does train train bus, he visits one diagonal, same if it's bus train train
otherwise 2 diagonals
whole geode
I don't see how that works. Surely, TTB would visit two diagonals.
long stirrup
why
whole geode
Like, you go from 1 to 18, then 18 to 6, then 6 to 7
whole geode
Is it?
I don't know
Like because it's a odd-gon there is no single longest diagonal, but maybe there's just the two
long stirrup
yeah, there are 2 long diagonals either way
and a lot of short ones
we must assume long ones then
aesthetically
whole geode
If we assume long ones only then there are many unreachable routes
long stirrup
we can also assume he never goes back, he always chooses the other diagonal
although then it's always two diagonals
long fog
Trains rule is any diagonal
whole geode
Assuming short diagonals, wlog, we start at station 1. Now does the order of tickets matter? There are three pertinent possibilities, the route involves station 1, the route involves station 2 or 23, and the route involves none of 1, 2, or 23.
Case A, the route involves station 1, let's say it's 1-n. Then T?? has a possibility of picking the correct route. BTT also does but we would have to go to station n from the first train.
We can see that we get two chances in the case of TBT if we are off by 1 on the first train, take the bus to n, then take the train back to 1.
So the order does matter, seemingly for case A.
long fog
Order of tickets does matter
whole geode
I mean, yes, I just showed that.
And I think this has gotten far more complicated than I want to do. Either there is a slick method for simplifying this that I'm not seeing, or this question is an absolute slog of case work
long fog
it is :/
long stirrup
the order doesn;t matter if you never go back
long stirrup
you visit exactly 2 diagonals, it's just a choice between 2/23 or 2/230
whole geode
Some situations you get two chances
Some you only get one
The choice of diagonals is not uniform
long stirrup
i don't understand 🤷♂️
whole geode
Assuming short diagonals, wlog, we start at station 1. Now does the order of tic...
I explained it here
long stirrup
oh you're assuming the notes are at the station, someone found them and returned
i;m assuming the notes are under the seat
and that there's one train per route
whole geode
@long stirrup an example might help if I wasn't clear enough.
Let's say the route is 1-7.
If we choose TTB, then there is a 1/21 chance we pick the 1-7 route. But if we pick any other route we fail, because if we go to (for instance) station 21, we have no way of accessing the correct train from this state.
If we do TBT, then we have the same 1/21 chance of picking the correct route. But then if we go to station 6, we have a 1/2 of going to station 7, then another chance of going to station 1 from 7 and getting on the right diagonal.
On the other hand for BTT, we have a 1/21 chance of going to 7 and a 1/21 chance of going to 1.
long stirrup
thank you
whole geode
So if the route contains station 1, BTT has a (1/21)^2 chance, TTB has a 1/21 chance, and TBT has a 1/21 + (2/23 * 1/2 * 1/21) = 1/21 + (1/21)^2 chance
If the route contains 2 or 23 then the probabilities are different as well
,w 23 choose 2
thorny flameBOT
whole geode
Oh Lord ok there is 4 cases.
There's a secret hidden case where the route is exactly 2-23
long stirrup
i think it's uniform
whole geode
@long fog I'm going to stop here. But all you need is to work the cases, and combine them by weighting them by the possibilities of being in the case.
long stirrup
i would instead guess 2/23
whole geode
It might be that it simplifies nicely, but it's certainly not clear to me right now that it does
long stirrup
ok you're probably right, the long diagonals are different from the short ones
long fog
The ans is 2/345
whole geode
long stirrup
yeah, if it's short, you get some kind of chance to visit it
and if it's long you get a different kind
and maybe if it's very short there's a third kind
whole geode
The ans is 2/345
That simplified nicer than I expected it would tbh
345 is a bit of a mystery to me though
,w prime factors 345
thorny flameBOT
whole geode
Anyway, for real going afk
long stirrup
i'm still on it
topaz sinewBOT
@long fog Has your question been resolved?
weak needle
idk if this is the right place to ask, but i want to teach myself multivariate calc what textbook should i get
topaz sinewBOT
idk if this is the right place to ask, but i want to teach myself multivariate c...
haughty wren
#book-recommendations maybe
vapid hawk
I did a course recently. Had Calculus by Strauss. Might help...
weak needle
okay, thank u guys
didnt see theres a section for book recommendations
my bad
.close
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mortal sphinx
Hey
wooden moon
b) is Vector AE = 1/2 AB + 1/2 BC
mortal sphinx
ok and c)
wooden moon
c) BC = 2EC - 2AF
because 2AF is AB
2EC is AC
AB + BC = AC, triangular vector addition law
so BC = AC - AB
BC = 2EC - 2AF
mortal sphinx
Thank you
Add the following vectors using trignometry (i.e cosine and sine laws) a) 7 m/s [N 30 E] and 2 m/s [S 17 E]
topaz sinewBOT
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haughty ember
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serene atlas
What is required for writing a equation Of line?
haughty ember
Im not sure what you mean, but I assume its just parallel and goes through (-2, 3)
serene atlas
I meant
Slope and y intersept right?
What does being parallel to y=4x-3 tells you about the slope of line we want.
haughty ember
Tells me that the slope of the line is also 4x
haughty ember
Slope and y intersept right?
Yeah
Nevermind, I used point slope and got y=4x+11
Is that correct by chance?
serene atlas
Yeah
topaz sinewBOT
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serene quiver
topaz sinewBOT
serene quiver
Why is it not acceptable to have the bounds of integration for "z" be from 0 to 16?
When radius = 0, z = 16
The answers do not match, so I don't really understand the logic for why this is wrong.
My idea:
The right answer:
I checked the 3D graph. The z only goes to 16. So why is bound from 0 to 16 wrong?
frail reef
Then you would be integrating the volume up to the plane z = 16, not the paraboloid z = 16 - x^2 - y^2
If we just imagine the integrand as 1 instead of x + y + z for a sec
serene quiver
Oh, I see.
So we are finding the solid under the paraboloid, whereas integrating from 0 to 16 would be integrating under the plane z=16, not under the paraboloid where z = 16 - r^2.
That makes sense. Thank you!
(unless i got that wrong...?)
frail reef
Oh, I see.
So we are finding the solid under the paraboloid, whereas integratin...
Basically yeah
topaz sinewBOT
frail reef
Calculating the volume of the cuboid will be different to calculating the volume of whatever shape you call that paraboloid
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clear wing
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static viper
Do you know how to find a tangent of a equation?
frail reef
You would find the gradient at that point
And you need to know a point where the line goes through
Then you can use that formula. I think it’s called the point slope formula or smth
clear wing
Do you know how to find a tangent of a equation?
can you guide me according to this method
<@&286206848099549185>
ivory wave
topaz sinewBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
clear wing
ya i do
but i am try to do it with other way
x = xx1 and y = yy1 xy = xy1 + yx1 / 2
this one
any idea how to do this
<@&286206848099549185>
<@&286206848099549185>
topaz sinewBOT
@clear wing Has your question been resolved?
clear wing
<@&286206848099549185>
topaz sinewBOT
@clear wing Has your question been resolved?
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neon iron
!occupy
neon iron
.occupy
so...
we're taught integration, differentiation, by our physics teacher... but don't have these topic this year and I wanted to have a good grip in these topics
and since we'll be taught about these topics by next year... so I'm not daring to open books of the next year
but as I want to be better in these tough sections
meager dawn
here are two external resources, the second of which has practice problems
neon iron
to practice well
neon iron
what school grade are u in rn?
11th grade
jade thunder
https://youtube.com/playlist?list=PLF797E961509B4EB5&si=RBPZF9VuF3Y5fEEP
https:...
This guy is hella good
neon iron
yea well im sure u a jee aspirant 😹
rigid ivy
to practice well
Khan Academy is the most popular recommendation in here
jade thunder
I’m a big fan of Leonard
neon iron
yea well im sure u a jee aspirant 😹
guessed it right 😂
thanks guys, means a lot
.close
topaz sinewBOT
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winter wave
how to solve this? the answer is 7/18. please help 🙂
topaz sinewBOT
how to solve this? the answer is 7/18. please help 🙂
opal vault
work backwards
winter wave
sorry, i dont know the formula to do this 😦
opal vault
it's just about finding how to reach O
let's assume you're at position X after 3 dice rolls
how can the next dice roll bring you to O?
winter wave
score 6
winter wave
i dont know...
opal vault
think about it
maybe X can be a specific position
that can allow OTHER dice numbers to reach 0
winter wave
x can be -1, if it is 1 2 or 3 it reaches O
winter wave
but how can i find the number 7/18?
winter wave
ok
opal vault
the event "you're on O after 4th roll" can be summed up into :
- either you rolled 6,
- either you rolled 4,5 being on 1 after 3rd roll,
- either you rolled 1,2,3 being on -1 after 3rd roll
so
let's write $X_n$ the place you're at after the n-th roll
thorny flameBOT
rafilou2003
opal vault
(X_0 = 0)
winter wave
Ok
opal vault
- either you rolled 6,
- either you rolled 4,5 being on 1 after 3rd roll,
- eith...
keeping track of this, knowing the probabilities of rolling each dice number is 1/6:
opal vault
$P(X_4 = 0) = \frac 16 + \frac 13 P(X_3 = 1) + \frac 12 P(X_3 = -1)$
thorny flameBOT
rafilou2003
winter wave
exactly
opal vault
got it?
opal vault
1/6 + 2/6 + 3/6
well that would be 1
opal vault
?
opal vault
**rafilou2003**
there's nothing more than this for now
we have to find the missing probabilities
winter wave
Which one is missing?
opal vault
**rafilou2003**
we don't know what P(X_3 = 1) and P(X_3 = -1) is
opal vault
do the same kind of reasoning
suppose we're on X after 2 rolls
how do we get on 1 with the 3rd roll
and alternatively how do we get on -1 with the 3rd roll
opal vault
?
winter wave
If x=O end we want to get in 1
we need to get positive
if x=0 and we want -1 we need negative
opal vault
well you gotta take into account all possibilities
winter wave
if x = -1 we would need 2 positive rolls
winter wave
I dont know how
opal vault
what can X be and what should we roll
winter wave
in 2 rolls, X can be= 0, 1, 2 , -1 , -2
winter wave
0 or 2
0 if it is 1, 2 or 3
2 if it is 4 or 5
If we want to reach 1 in the 3rd roll
X must be on 0 or 2
correct?
winter wave
its needs to be -2 or 0
thorny flameBOT
rafilou2003
winter wave
How can i write it?
opal vault
You told me there's 2 ways for $X_3=1$
thorny flameBOT
rafilou2003
winter wave
to $P(X_3=1)$ we need 0 or 2
thorny flameBOT
zFxlipe
opal vault
Either $X_2 = 0$ and we roll 1,2,3
thorny flameBOT
rafilou2003
opal vault
Either $X_2 = 2$ and we roll 4,5
thorny flameBOT
rafilou2003
winter wave
Yep
opal vault
What's the probability that X_2 = 0 AND we roll 1,2,3?
winter wave
1/2
opal vault
No that's just the probability that we roll 1,2,3
opal vault
We want the probability of BOTH events happening
Since they're independent, it's the product of their probabilities
winter wave
1/6 two times or 1/2 + 1/3
plus the 1/2 of the X_3
If we roll 2 times, it need to fall on 6 two times and stay at 0; If we roll 2 times, it need to be 1 forward and 1 backward to stay at 0; to get X_3 to 1 2 or 3 we need 1/2
opal vault
?
I just wanted you to write the product of 1/2 with P(X_2=0)
$\frac 12 P(X_2 = 0)$
thorny flameBOT
rafilou2003
opal vault
What's the probability that X_2 = 0 AND we roll 1,2,3?
That's the prob
winter wave
How can I do it?
opal vault
What is this
opal vault
1/2 P(X_2 = 0)
winter wave
How did you get this number
winter wave
To achieve X_2=0 and roll 1,2,3
you have two rolls of the dice
the third roll would be 1,2 and 3
opal vault
Third roll would be 1 2 or 3
winter wave
Exactly
winter wave
Yep
opal vault
Meanwhile, probability that X_2 is 0 is just P(X_2 = 0)
winter wave
So X_1 can be = -1, 0 or 1
opal vault
So probability of both happening is 1/2 * P(X_2 = 0)
opal vault
We can focus on P(X_2 = 0) if you want
How can we get on 0 after 2 moves
(I think we can do 2 moves at the same time it should be easy)
winter wave
X_3 can be = -3, -2, -1, 0, 1, 2 ,3
X_4 to be 0 needs X_3 to be -3, -2, -1, 0, 1 ,2 ,3 if the dice rolls 6, or -1 and 1 if the dice rolls positive or negative
opal vault
Yep that's what we said earlier
We just kept it simple and said "either dice roll is 6, or it's not 6 and we're on -1 or 1"
opal vault
... that's what we've been going for this entire time
winter wave
-3, -2, -1, 0, 1 ,2 ,3 = 7/X
opal vault
No that's not how you do that
winter wave
-1 or 1 is suitable if the dice rolls positive or negative or 6, so we can add that
so it would be 7 out of 18
because the dice has 6 sides
we roll it 3 times
wouldnt it be 7/24?
opal vault
Again, you're just making numbers do whatever you want
winter wave
I am bad at math, i am sorry
opal vault
And you're not thinking about computing probabilities
winter wave
How can i compute probabilities?
opal vault
I wanted us to take a moment to look at P(X_2 = 0) for a moment
winter wave
ok
opal vault
We always start at 0
winter wave
yep
winter wave
the dice needs to achieve 6 both times, or achieve 1, 2 and 3 one time and 4,5 the other
opal vault
Almost
winter wave
Is there another way?
ohhhh
i get it
it can achieve 1,2,3 or 4,5 in the first and 6 on the second
opal vault
You don’t need the dice to achieve 6 BOTH times
winter wave
the important is the second one
opal vault
A 6 on the second roll would bring us back
winter wave
yep
winter wave
if its goes up and up, it would go to 2
if goes back two times, it would go -2
so the probability is half
opal vault
Ok I see your problem
winter wave
the chance of X_2 being 0 is 1/2, right?
opal vault
You're always thinking that the probability of going up and going down is the same
winter wave
You're always thinking that the probability of going up and going down is the sa...
oh yeah
opal vault
We're not doing coin tosses
opal vault
We're doing dice rolls
opal vault
Well just take into account the real probabilities
Say the first roll is +1 and the second roll is -1
What's the probability of that
winter wave
the first roll is 1/2 and the second 1/3
opal vault
No ok there's one thing that you gotta learn right now
winter wave
5/6?
winter wave
ok
opal vault
If you assume they're independent (meaning the outcomes dont affect each other)
Then P(A and B) = P(A)*P(B)
The probability that BOTH happen is the product
winter wave
Oh, how can I multiply it?
opal vault
Adding probabilities is for when you have SEPARATE events that can't happen together
winter wave
Ok
opal vault
P(A or B) = P(A) + P(B), when A and B can't happen together
winter wave
i understand it now
since they are happening together
we need to multiply them
so it would be 1/6
But then we still have the 6
so, how we achieve 1/2?
winter wave
How can I know?
opal vault
There we go
winter wave
Oh, ok
winter wave
I got it
thats why it would be 1/6 x3
we need to account these two together
ok
so, for $X_2=0$ we have 1/6 happening three times, resulting on 1/2
thorny flameBOT
zFxlipe
opal vault
Remember we were looking at this because we were looking for P(X_3 = 1)
And we said
That either we have X_2 = 0 and we roll 123 on third roll
Giving us $\frac 12 P(X_2 = 0)$
thorny flameBOT
rafilou2003
opal vault
Or we have X_2 = 2 and we roll 45
opal vault
Or we have X_2 = 2 and we roll 45
What would be the probability of that
winter wave
if we have X_3 = 1,
123 on the third roll would be 1/2
if X_2 = 2 we roll 45 it would be 1/3
opal vault
1/3 is the probability that we roll 45 ALONE
winter wave
to roll both of them together and get 45
opal vault
It's not the probability that we roll 45 AND THAT X_2 = 2
winter wave
how should I do it?
winter wave
two times getting 123
opal vault
Uhhuh
winter wave
1/2 times two?
winter wave
we need to multiply
opal vault
And the probability of getting it the 2nd time?
opal vault
on the second time it would be 1/4 together?
Yes 1/2 * 1/2
opal vault
Okok
winter wave
Yep
opal vault
I'll let you input the values we found
opal vault
1/2 * P(X_2 = 0)
What's the value of this
opal vault
Yes
opal vault
1/3 * P(X_2 = 2)
And this
opal vault
We just went through it
opal vault
1/3 x 1/4?
Yes
opal vault
Yes
winter wave
Ok
opal vault
So ? + ? = ?
winter wave
To get P (X_3 = 1) we need to multiply 1/4 x 1/12?
winter wave
So P (X_2 = ?)
yes
to X_2 = 0 we have 1/4
X_2 = 2 we have 1/12
so i add them together now?
+?
opal vault
Yes
opal vault
Yes
opal vault
Yes
opal vault
We went through it
winter wave
If they are together, multiply
if they are separete, add
ok
I get it
Ok
so we have P(X_3 = 1)
to get P(X_4 = 0) we need: 45 or 6
both being 1/3 and 1/6
should I multiply them?
1/18?
or 1/2?
But P can be any number on X_3
so, P(X_3 = -3, -2, -1 ,0 , 1 , 2 ,3)
?
opal vault
Well did we cover all cases?
opal vault
**rafilou2003**
We had this earlier
1/6 is easy
We did the P(X_3 = 1) = 1/3
So what about P(X_3 = -1)?
opal vault
For -1? No
winter wave
So what about P(X_3 = -1)?
It needs to be P(X_2) = -2 or p(X_2) = 0
opal vault
Yes
winter wave
x_2 0 is 1/2
opal vault
Yes
winter wave
should i multiply or add them
winter wave
-2
opal vault
Ok
winter wave
ohh ok
opal vault
And when do we need 45
winter wave
1/3?
opal vault
1/3 * 1/2
opal vault
No
winter wave
or 1/18 + 1/6
opal vault
Separate events
winter wave
ok
opal vault
Bad math
opal vault
Yes
thorny flameBOT
zFxlipe
opal vault
P(X_3 = -1) = 2/9
winter wave
yep
opal vault
P(X_3 = 1) = 1/3
winter wave
now, shal we add each other?
opal vault
**rafilou2003**
Ill let you finish
winter wave
1/6 + 1/3 + 1/2?
opal vault
No
winter wave
how should i go on now?
opal vault
You literally have all data
if x = 4 and y = 5, what is 2x + y?
This is the same kind of thing
winter wave
13
opal vault
So
opal vault
It's 1/6 + 1/3 * (first prob we found) + 1/2 * (second prob we found)
It's literally written here
winter wave
Ok
opal vault
**rafilou2003**
Here
winter wave
which order should i do?
winter wave
wdym?
opal vault
What is P(X_3 = 1)?
winter wave
1/3
opal vault
What is P(X_3 = -1)?
winter wave
2/9
opal vault
:_:
opal vault
What is 1/6 + 1/3 * P(X_3 = 1) + 1/2 * P(X_3 = -1)?
So what's this?
winter wave
Thats X_4 = 0
opal vault
No the value