because of that you multiplied by 3! before
#help-13
tacit citrus
yhea
unreal shale
I believe one shouldn't divide by two, but it all depends on what the teacher meant, so I think you should ask them
tacit citrus
my exam is on wednesday
and i just have classes tuesday
:(((
but yhea
i ill try to ask my teacher
unreal shale
If the order of the boxes matters because they are different, then we shouldn't divide by 2! because (3a,2b,2c) is not the same as (3a,2c,2b) ; but, if the order doesn't matter, then you should divide by 2!, because those would be considered a repeated case
tacit citrus
ok
but for example
in the exercise that i showed u
when they did that multiplication of C(8,2)xC(6,2)xC(4,2)
they had to divide by 3! because it looks like
that by doing that multiplication some kind of order was already there
becasue if you see
when they do C(8,3)xC(5,2)
they just dont do nothing
unreal shale
they divide by three because if the same photos that go in envelopes a, b, and c, go into another envelope, the combination would be considered the same, and the calculation you did computes all those cases. We have 3! ways of disarranging those 2 photos in a, 2 in b, 2 in c, so that's why we divide everything by 3!; so that we are left with only 1 of those cases (remember the last two envelopes are fixed to have just one photo, and since the order of the envelopes don't matter, we don't have to multiply nothing more)
in the boxes case, boxes are all different, so when computing for the 3,2,2 cases we'll have:
7C2 • 4C2 • 2C2 : for all the possible ways of choosing balls for the boxes.
Lets say balls are called 1,2,3,4,5,6,7.
Cases would look like (123, 45, 67) where a has 123, b has 45, and c has 67
What we computed above will give us all the possible cases, so for example: (457, 12, 36), (345, 67, 12), etc.
Since we used binomial terms (7C2, 4C2...) we already considered the fact that (123,45,67) = (321,54,76) so we don't have to worry about them being repeated. Multipliying Binomials this way will only consider (123,45,67), so if we want to consider, let's say, (45, 123, 67) we will have to find the permutations of those 3 numbers. In the envelopes case, since we didn't care, we would say that those two cases were the same so we wouldn't need to multiply by anything.
Boxes are different. This means that if the same balls in b are in c, and viceversa, the set is not the same. Because of that, we will have to consider those extra 3! permutations.
Finally, the combinations before altrady had calculated (123,67,45) and (123,45,67), so when we multiplied by 3! these are getting counted again. We will have to divide by 2! if the order of the boxes is not considered.
Now that i read it again, I think I agree with you, you should divide by 2! because they don't really ask you for the order of the boxes, they just say they are different. If in any problem they tell you that the order matters, you should analyse if those cases had been counted or not. I'm sorry for the confusion 😔
unreal shale
Thanks you for listening and sorry for not reading well 😭 I hope i did more good than bad
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crimson sedge
Okay I have a question I amnew
cedar kilnBOT
Okay I have a question I amnew
crimson sedge
anyone willing to help
flint plinth
just ask your question, you don't have to ask to ask
cedar kilnBOT
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crimson sedge
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crimson sedge
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@crimson sedge Has your question been resolved?
crimson sedge
solve it?
mild oyster
which one do you need help with
crimson sedge
16
crimson sedge
no help?
cedar kilnBOT
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Available help channel!
vapid basalt
help
loud gyro
ask after 7th. No cheating in this server
vapid basalt
these are easy marks
loud gyro
This looks like graded exercise
vapid basalt
ask after 7th. No cheating in this server
after 7 pm?
loud gyro
7th november
vapid basalt
ask after 7th. No cheating in this server
i'm in college
please can you
this contain 10 marks
and i don't wanna lose it
vapid basalt
7th november
west path
no
cedar kilnBOT
@vapid basalt Has your question been resolved?
arctic dock
Can someone help me solve for 3 idk what to don
cedar kilnBOT
Can someone help me solve for 3 idk what to don
pastel vault
Can someone help me solve for 3 idk what to don
do you mean question 2 or 3?
assuming you mean question 2
keep going, evaluate the integral you get at x = b and x = 0
then you get a quadratic equation in b
cedar kilnBOT
@arctic dock Has your question been resolved?
arctic dock
then you get a quadratic equation in b
Do I just use quadratic formula after
pastel vault
Do I just use quadratic formula after
yeah
pastel vault
,w integrate (2+6x-3x^2) from 0 to b
wraith daggerBOT
pastel vault
,w (-b^2+3b+2)=3
wraith daggerBOT
pastel vault
yeah it's not factorable in the rational numbers
arctic dock
Oh ok
arctic dock
yeah it's not factorable in the rational numbers
Oki I got this would it factor down to 2+3b-b^2
pastel vault
yeah in fact for all definite integrals, you will have (+c) - (+c) = 0
so you do not need to write +c for a definite integral
pastel vault
Oki I got this would it factor down to 2+3b-b^2
yep
arctic dock
ok so I just factor that out but do i keep =3 at the end
pastel vault
ok so I just factor that out but do i keep =3 at the end
yes
you make the RHS zero
and then use the quadratic formula
arctic dock
what happens to the 1/b in the front
pastel vault
Oki I got this would it factor down to 2+3b-b^2
you already divided by b to get here
wooden vortex
wooden vortex
<@&286206848099549185>
cedar kilnBOT
@wooden vortex Has your question been resolved?
fair nymph
cedar kilnBOT
fair nymph
Can someone explain how to do these type of questions
humble karma
You're given the substitution for all of them, I take it that you've seen u-substitutions?
cedar kilnBOT
@fair nymph Has your question been resolved?
shut magnet
I need help with the following question:
let {an} be an infinite series such that:
- an≠0 for every n
- 1/an doesn't have any bounded subsequences.
prove lim(an)=0
slate lintel
if 1/an doesn't have any bounded subsequences then every subsequence will eventually be arbitrarily big
shut magnet
if 1/an doesn't have any bounded subsequences then every subsequence will eventu...
I know but I need a formal proof🥲
slate lintel
you're right, you do
but to make a formal proof you should understand why it's true
what does it mean for lim an to be 0?
shut magnet
I have an idea, can someone check it:
assume lim(an)≠0
then there is e>0 such that for every N there exists n>N such that |an|≥e
=> |1/an|≤1/e
so if we pick those as our subsequence we get a bounded subsequence, contradiction.
slate lintel
yep that's pretty much what I had in my head
shut magnet
yippeeeee
slate lintel
if it keeps escaping the box then that's a bounded subsequence
so eventually it must stop escaping the box
shut magnet
I understand the topic well so I can easily imagine those things in my head my only problem is when I need to formally prove those things
shut magnet
anyways thanks for the virtual brainstorming😅
vivid karma
Hi i have a question. When do you usually use U-substitution?
vivid karma
I'm very confused because the upcoming test doesn't say when I should use it. I'm scared that if I use the substitution then ill get a different answer
like normal integration vs u-sub
bleak viper
Hi i have a question. When do you usually use U-substitution?
when there is composition of functions and there is some relation (derivative) between those functions
vivid karma
could you explain a bit simpler 😭
bleak viper
for functions that can be written as product of a function (u) and its derivative (du)
,, \int{udu}
wraith daggerBOT
Astar777
vivid karma
i see
bleak viper
lets say you have $\int{x^2 2x dx}$
wraith daggerBOT
Astar777
bleak viper
(u can do this without u-sub, but im just showing example)
you can let x^2 = u so 2xdx = du
substituting it, you get $\int{udu}$
wraith daggerBOT
Astar777
bleak viper
you can do the same u-sub in this one too, $\int{\cos{(x^2)}2xdx}$
wraith daggerBOT
Astar777
vivid karma
so would there be a question where normal integration and u-substitution answers are different
idk if im doing it right
supposedly if i use u-substitution i should get t his answer
but i tried using the normal integration method and i got a different one
bleak viper
if u expand the u-sub answer, you'll get the answer u got from normal integration
bleak viper
yes
vivid karma
okay thank u so much!!!!!
bleak viper
both are same
vivid karma
that really helped me
oh last question sry
whats the point of using u-substitution over normal integration
or other way a round
im having an easier time using u-sub
can i just use that for everything?
bleak viper
you use u-sub to make it easier to integrate
vivid karma
i see
wraith daggerBOT
Astar777
vivid karma
so i would do u = x^2?
bleak viper
yes
you get cos(u)du, which is very simple to integrate compared to the original integral
vivid karma
im also having trouble with the cos sin tan
thing
i just have to remember i guess
idk what to do after that]
bleak viper
whats the integration of cosx
bleak viper
sinx
bleak viper
yes, just replace u with x^2
bleak viper
bleak viper
happy to help
cedar kilnBOT
If you are done with this channel, please mark your problem as solved by typing .close
vivid karma
.close
crimson sedge
Quick question: How would you read this?
cedar kilnBOT
Quick question: How would you read this?
south tundra
If ~ is an equivalence relation, presumably "Maximal subset C of V such that u is equivalent to v for all u and v in C"
crimson sedge
~ means that if there is a path u to v, then there is also a path v to u
I'm not sure if that's that equivalence relation means
mental trail
~ means that if there is a path u to v, then there is also a path v to u
do you mean 'u ~ v means there is a path from u to v'
and properties are
that u ~ v is the same as v ~ u
u ~ u
crimson sedge
do you mean 'u ~ v means there is a path from u to v'
Yes
mental trail
and u ~ v and v ~ w imply u ~ w
crimson sedge
Wait lemme show you
mental trail
those three properties
south tundra
Then "Maximal subset C of V such that there is a path from u to v and from v to u or u = v for all u and v in C"
mental trail
make an equivalence relation
crimson sedge
mental trail
ok i get it
crimson sedge
Then "Maximal subset C of V such that there is a path from u to v and from v to ...
So there is no "short" way of saying ~ in this case?
crimson sedge
So if you were reading that sentence out loud, how would you say it?
mental trail
So there is no "short" way of saying ~ in this case?
if you assume that there exists a path from u to itself (the path where you don't move) you can make it shorter
A strongly connected component is a maximal subset of the set of vertices where every vertex has a path to every other vertex
crimson sedge
That makes sense
mental trail
those "strongly connected components" are exactly the subsets $[u]_\equiv$
wraith daggerBOT
rafilou is not not born in 2003
mental trail
where $[u]_\equiv$ denotes the set of all vertices that have a path to u and u has a path to
wraith daggerBOT
rafilou is not not born in 2003
cedar kilnBOT
cedar kilnBOT
Available help channel!
cedar kilnBOT
south tundra
King's rule:
[ \int_a^bg(x)\dd x = \int_a^bg(a+b-x)\dd x ]
wraith daggerBOT
A Loner Bean
eager hemlock
I know that it's kings rule, but how?
mighty drift
this is an expression, wdym "prove this"
south tundra
I know that it's kings rule, but how?
Apply it to to the integral on the left, expand the parenthesis, see what equation you get (namely, try to solve for the integral on the left)
mighty drift
oh
The low bar of = is super thin, it didn't display
eager hemlock
lolll
wheat ledge
King and add rule
eager hemlock
uhh i already said i knew ..
wheat ledge
👍
cedar kilnBOT
eager hemlock
oops
eager hemlock
King's rule:
\[ \int_a^bg(x)\dd x = \int_a^bg(a+b-x)\dd x \]
deleted the original question mistakely, but could you check that if im doing correct
cuz I just really don't know where's wrong
cedar kilnBOT
Available help channel!
south tundra
What you wrote is correct but you are making this more complicated than necessary, there's no need to consider the integral from 0 to pi, start by simply applying the king's rule to the integral of xf(sinx) from 0 to pi/2
eager hemlock
What you wrote is correct but you are making this more complicated than necessar...
but the orgianl question's integral is all from 0 to pi
south tundra
My memory must have deceived me then
south tundra
Ah, it's from 0 to pi on both sides
eager hemlock
yesyes
south tundra
Start by applying the king's rule to the integral of xf(sinx) from 0 to pi
eager hemlock
Start by applying the king's rule to the integral of xf(sinx) from 0 to pi
but why the method above is wrong
south tundra
but why the method above is wrong<:cat_happycry:636407813144576010>
It's not wrong like I said
south tundra
Your rhs is
[ \pi\int_0^{\frac{\pi}2}f(\sin x)\dd x ]
The rhs in the problem is
[ \frac{\pi}2\int_0^\pi f(\sin x)\dd x ]
Why are these different?
wraith daggerBOT
A Loner Bean
eager hemlock
$\int_0^{\pi}f(\sin x)\dd x=2 \int_0^{\frac{\pi}2}f(\sin x)\dd x$
wraith daggerBOT
Allophane
sacred grail
you couldve done this much more efficiently using what bean said from the start
eager hemlock
you couldve done this much more efficiently using what bean said from the start
Yes I've done it, just wonder why this way is not inconsistent with that
.close
oops
south tundra
.close
vernal kite
is the working out for 2a correct?
cedar kilnBOT
is the working out for 2a correct?
vernal kite
am i still able to apply the rule of lim k * f(x) = k * lim f(x)?
alright thanks
.close
cedar kilnBOT
vernal kite
.reopen
cedar kilnBOT
✅
vagrant elbow
yes
vernal kite
okay
can someone tell me how you'd do 1a for level 2?
is cos4theta the same as cos^4 theta?
frank solar
is cos4theta the same as cos^4 theta?
No
vernal kite
so how do i simplify 1-cos4theta?
frank solar
I believe you have to use the limit $\lim_{x \to 0} \frac{1-\cos(x)}{x^2}=\frac{1}{2}$
wraith daggerBOT
cristorenzo99
vernal kite
but 1-cosx doesnt equal anything
frank solar
**cristorenzo99**
Do you know this special limit?
vernal kite
nope
frank solar
You know $\lim_{x \to 0} \frac{\sin(x)}{x}=1$ I guess, no?
wraith daggerBOT
cristorenzo99
vernal kite
yes i do
frank solar
Maybe you have to use that $\cos(2x) = \cos^2(x)-\sin^2(x)=1-2\sin^2(x)$
wraith daggerBOT
cristorenzo99
frank solar
$4\theta = 2(2\theta)$ and you can use the formula with $x=2\theta$
wraith daggerBOT
cristorenzo99
vernal kite
oh so 2(1-2sin^2(x))?
frank solar
Using the formula above, $\cos(4\theta) = \cos^2(2\theta)-\sin^2(2\theta) = 1-2\sin^2(2\theta)$
vernal kite
oh so cos4theta still equals 1-2sin^2theta
wraith daggerBOT
cristorenzo99
frank solar
I missed a two! Sorry!
hot crag
I missed a two! Sorry!
if you edit your original tex message, the tex changes
vernal kite
what the skibidi am i doing
can u guide me through the process of simplifying this
so i got 1-1-2sin^2(2theta) / theta^2
correct?
frank solar
Have you reached where the initial limit is equal to $\lim_{\theta \to 0} \frac{2\sin^2(2\theta)}{\theta^2}$?
wraith daggerBOT
cristorenzo99
frank solar
oh wait is it 1-(1-2sin^2(2theta))?
That's equal to the numerator I wrote
frank solar
Now write $\frac{\sin^2(2\theta)}{\theta^2}=\frac{\sin(2\theta)}{\theta}\frac{\sin(2\theta)}{\theta}$
wraith daggerBOT
cristorenzo99
frank solar
And use the limit of sin(x)/x (adjusting the terms!)
vernal kite
isn't that 4?
frank solar
I believe it's 8
vernal kite
wait whaat
frank solar
**cristorenzo99**
There is a two in front of the fraction
frank solar
w, lim_{x ->0} (1-cos(4x))/x^2
frank solar
w, lim_{x ->0} (1-cos(4x))/x^2
I don't remember how to use wolfram alpha here
frank solar
silly question but why doesn't sin2theta x sin2theta = sin^2 4 theta?
$\sin(2\theta) \cdot\sin(2\theta) = \sin^2(2\theta)$
wraith daggerBOT
cristorenzo99
vernal kite
yea but whats the reasoning behind it
azure stirrup
yea but whats the reasoning behind it
u can think of sin(2θ) as a variable like x or y or k
so yk how x times x = x^2
it’s essentially the same idea
sin(2θ) times sin(2θ) = sin^2 (2θ)
azure stirrup
sheesh u go dr du
vernal kite
yea i know]
azure stirrup
that helps thanks
nws
vernal kite
for 2b, i’m getting 1911 but the answer is 1910
what am i doing wrong
same as 2a
it's a second larger
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
rustic raven
For the relation in example 8, is it wrong to say x,y : y=+ or - sqroot of x?
rustic raven
In set builder form
solid juniper
your answer is missing some details
rustic raven
How?
Do u say such that x belongs to P and y belongs to Q? I will add that detail anyway
solid juniper
yes that’s the main thing missing
rustic raven
Other than that
rustic raven
For the relation in example 8, is it wrong to say x,y : y=+ or - sqroot of x?
Here it says x is the square of y
That's why i asked whether my way was correct
worldly chasm
It's preferable to use the square over the square root because every real number has a square but not every real number has a square root that is also a real number
So then there is a question about domain and so on
rustic raven
It's preferable to use the square over the square root because every real number...
But I mentioned that x belongs to P
solid juniper
yea like its fine but idk why you would prefer sqrt here
worldly chasm
Do complex numbers exist in Q?
rustic raven
yea like its fine but idk why you would prefer sqrt here
Just came to my mind, wanted to know if it was correct
rustic raven
Do complex numbers exist in Q?
No
worldly chasm
How do you know?
solid juniper
P and Q are the sets in the figure
rustic raven
How do you know?
Q is just 5 3 2 1 -2 -3 -5
worldly chasm
Oh, the P and Q are explicitly and completely named
I thought it was just a sample
rustic raven
Yes
worldly chasm
Fair enough
rustic raven
For the relation in example 8, is it wrong to say x,y : y=+ or - sqroot of x?
Also another question about the "note"
Does it have something to do with power set? Because I remember reading that for a set of n elements there are exactly 2^n elements in its subset
worldly chasm
Yes exactly
solid juniper
yea
rustic raven
Oh really?
worldly chasm
Yeah, a relation is just some set of ordered pairs of elements
rustic raven
So all relations are subsets of the cartesian product
worldly chasm
And each ordered pair you can choose to either exist or not exist
rustic raven
Ok
worldly chasm
There are pq ordered pairs, and 2 possibilities for each
worldly chasm
Yes
rustic raven
What
worldly chasm
Same with power sets
rustic raven
is it related to permutations and combinations?
worldly chasm
It is related to the sum of the different ways to do a combination on n elements
rustic raven
Oh
worldly chasm
$\sum_{k=0}^n \binom{n}{k} = 2^n$
wraith daggerBOT
OmnipotentEntity
rustic raven
OH that nCr thing
Alright got it
This notation is werid
Alright thanks for the help
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
modest mason
im reallo confused how im they differentiate the stuff inside tthe function. i converted it to (2-x)(1+x)^-1 so i could use product rule to differentiate but my answer is completely off from the mark scheme. the rest of the question i understand how to do.
modest mason
im familiar that u have to use quotient rule to differentiate fractions like that although i just want to know where i was wrong when using product rule since im much more comfortable using that
livid hound
your power is incorrect
modest mason
omg r u serious
wait js realised i forgot the negative in the thing
but i did include negative in my working out
livid hound
show your work
livid hound
seems you forgot about the presence of arctan
royal canopy
so i am learning real analysis using ocw and rudin, and am stumped on how to prove that (a^n)^(1/m)=(a^p)^(1/q) if n,p>=0 if n/m=p/q
cedar kilnBOT
so i am learning real analysis using ocw and rudin, and am stumped on how to pro...
@royal canopy Has your question been resolved?
crimson sedge
haughty robin
cedar kilnBOT
haughty robin
How do i go about this question?
crimson sedge
Sorry Monique
haughty robin
O u beat me
crimson sedge
Lol yh mb
hot crag
notice you can rewrite that as
wraith daggerBOT
ren
hot crag
now there is some very simple rearranging, where you can get all the y terms on one side and the x terms on the other
crimson sedge
now there is some very simple rearranging, where you can get all the y terms on ...
Yes I understand
Then I can differentiate both sides respectively
hot crag
good. what will be the rearranged form?
crimson sedge
Oh yh that's what I meant mb'
crimson sedge
good. what will be the rearranged form?
One sec
crimson sedge
Divide by dy I see it
crimson sedge
Then flip after no?
wraith daggerBOT
ren
hot crag
now?
sick bough
just dropping in to say I admire your insistence for the d
crimson sedge
Huh
hot crag
thanks lol
crimson sedge
hot crag
yeah great job
hot crag
mhm
crimson sedge
Then use parameters given
crimson sedge
Ok
wraith daggerBOT
ren
crimson sedge
Sub in x=2 and y=3
crimson sedge
Ln2
hot crag
yep
crimson sedge
Bc ln2 - ln1
crimson sedge
3-5
hot crag
where lmao
hot crag
yeah that's assumed
crimson sedge
Assumed that it is always positive?
hot crag
no, modulus is assumed
crimson sedge
OK=k
crimson sedge
So ln(y - 5) = ln(x-1) + ln2
Then ln(y-5) = ln(2x-2)
Then y-5 = 2x-2
Bc raise both
To e
So y = 2x + 3
@hot cragIs this right?
If so
Do you reckon the best way to approach these questions is to change all y' terms to dy/dx and y'' to dy/dx^2
cedar kilnBOT
@crimson sedge Has your question been resolved?
cedar kilnBOT
crimson sedge
Anybody here?
random dawn
yo
dire geode
try using similar triangles
dire geode
oh then what exactly is the question because it's unclear from your diagram
crimson sedge
i have to find ?
dire geode
what does 90 cm refer to
dire geode
what does ? cm refer to
crimson sedge
so the blue thing is water, the 90 cm is the height of a barrel, the 50 cm is the diameter.
?cm refers to the height of the water
i found the english version
full notch
cedar kilnBOT
smoky finch
i can see a triangle forming
cedar kilnBOT
@full notch Has your question been resolved?
full notch
i can see a triangle forming
💀💀💀
west path
i mean its probably the way to go tbh
smoky finch
there must be some kind of property that will be used
empty kindle
Wow so perceptive
smoky finch
Wow so perceptive
was that in a sarcastic way?
full notch
i mean its probably the way to go tbh
I used it didn't work
smoky finch
what's the question?
empty kindle
I think a little bashing can work, lemme doodle something
I think I got it
but my drawing sucks
full notch
I've been sitting on these for a whole day now
empty kindle
lemme try to draw better
oh shit the colours lmao
anyways, so we find x is 4sqrt(3) then we can do the following:
empty kindle
what
smoky finch
you really cooked
full notch
He did in fact cook
cedar kilnBOT
@full notch Has your question been resolved?
compact epoch
If a TB X-ray examination produces a positive result, the person concerned is faced with the question of their chances of still being healthy. Use the term conditional probability to give an answer under the following three model assumptions:
(i) 0.1% of the population has TB.
(ii) In the case of TB carriers, the examination provides a positive result, i.e. a correct result, in 94% of all cases.
(iii) In healthy people, the examination produces a positive result in 1% of all cases, i.e. an incorrect result (false positive).
are you guys familiar with these kind of examples?
empty kindle
```
If a TB X-ray examination produces a positive result, the person concerned i...
what is the question?
compact epoch
what is the question?
Use the term conditional probability to give an answer under the following three model assumptions:
So there is a formula for this i think
Bayes theorem?
empty kindle
oh the question is the probability of the person being healthy given the x-ray was positive right?
so P(Healthy | Xray positive), by bayes theorem that would be P(Xray positive | Healthy) * P(Healthy) / P(Xray positive) = 1% * 99.9% / P(Xray positive)
now gotta calculate that last part
P(Xray positive) = P((Healthy and positive) U (Not healthy and positive))
since there is no intersection between those two events(someone can't be healthy and not healthy at the same time) this is = P(Healthy and positive) + P(Not healthy and positive)
P(Healthy and positive) = P(Healthy) * P(Positive | Healthy) = 99.9% * 1%
P(Not healthy and positive) = P(Not healthy) * P(Positive | Not healthy) = 0.1% * 94%
compact epoch
so P(Healthy | Xray positive), by bayes theorem that would be P(Xray positive | ...
do you know this formula by heart?
compact epoch
its actually intuitive if you draw venn diagrams
Ah interesting i will study the formula then
empty kindle
but anyways, finishing the question
compact epoch
here is the solution of out of the math workbook
empty kindle
we have P(Xray positive) = 99.9% * 1% + 0.1% * 94%
compact epoch
c on top is the complement i assume
empty kindle
yeah that's what I'm doing
compact epoch
yeah that's what I'm doing
So how did you know you had to use this formula? Is it having 2 probabilities ?
empty kindle
well, interpreting the text I know its a conditional probability: chances of being healthy given the xray was positive, and that probability is not given in the text(otherwise it wouldn't be a question lol) but I have the probability of the xray being positive given you are healthy, bayes theorem lets you switch them up
P(A | B) = P(B | A) * P(A) / P(B)
I want P(A | B), but I have P(B | A) and P(A), so I might as well use the formula
then you just have to calculate P(B)
compact epoch
I want P(A | B), but I have P(B | A) and P(A), so I might as well use the formul...
Ohh interesting. yeah i have to look into the transitions, but it looks like it is simple once you know it
empty kindle
actually
here
this pretty much sums up what you need to know most of the time
and we used almost every formula here
compact epoch
Oh how do you know if something is dependent or independent?
empty kindle
if I roll a dice today and one tomorrow, are the faces they will land dependent?
does the face the first dice got in any way change the probabilities for the second dice?
empty kindle
P(Healthy and positive) = P(Healthy) * P(Positive | Healthy) = 99.9% * 1%
P(Not ...
is this dependent or independent
compact epoch
What would be a case for dependent or partially independent
compact epoch
P(Healthy and positive) = P(Healthy) * P(Positive | Healthy) = 99.9% * 1%
P(Not ...
I have seen this thing in a lecture i think they are dependent
empty kindle
you're not supposed to recall the answer from somewhere
this is text interpretation
does being healthy influence the probability of the xray being positive
or vice versa, does the xray being positive influence the probability of you being healthy
empty kindle
in this case it clearly does, so we use the formula for dependent
compact epoch
Yeah it does have a big influence on those tests
empty kindle
one sec lemme draw something
suppose the rectangle is all the possibilities
the blue circle is all the cases where someone is healthy
the red circle all the cases where someone has a positive xray
traditionally the probability P(A) = number(A) / number(Omega)
where Omega is the set of all possibilities(the rectangle) and A is the set of the ones you're interested in
compact epoch
Yep
empty kindle
but when you have a conditional, that is, you know something extra, P(A | B) becomes number(A intersection B) / number(B)
since you know B is true that reduces the space of possibilities from omega to B
and cuts away everything inside A that is not also inside B
empty kindle
yeah
bad drawing but anyways, you're looking for the green area inside the red circle, not the blue area inside the black rectangle
compact epoch
Yep
P(A|B) , P(B|A) What is the difference here
I see them mixed up in these examples
empty kindle
depends on what A and B are
in our case A = Healthy, B = XRAY positive
so "Chances of being healthy given the xray was positive" vs "Chances of xray being positive given you are healthy"
compact epoch
But what does switching sides do
empty kindle
But what does switching sides do
it does bayes theorem
compact epoch
so "Chances of being healthy given the xray was positive" vs "Chances of xray be...
Oh damn
empty kindle
P(A | B) = P(B | A) * P(A) / P(B), algebrically thats what happens
and for what it means that's the message I sent before
btw if that's solved remember to .close the channel
or you have another question?
compact epoch
btw if that's solved remember to .close the channel
Ah ok yeah, but can i ask you one more?
empty kindle
ok
compact epoch
a) pairwise independent
b) completely independent
are. Be there
A the event "same sides in the last two throws"
B the event "same sides on the 1st and 3rd throw" and
C the event "same sides on the first two throws"
empty kindle
the question being you have to calculate the probabilities or what?
oh mb I read it incorretly
empty kindle
referring back to here
there is the intuitive sense in which something is independent or dependent
in the mathematics that reflects by whether P(A inter B) = P(A) * P(B | A) or P(A) * P(B)
if its just P(A) * P(B) then they are independent
so one thing you can do is calculate the chances of each of those events and check above whether it fits the formula for dependent or independent
compact epoch
Ah ok so just testing it can give us the answer
empty kindle
also, if you're gonna calculate the chance to check against the formula, obviously you can't just use the formula, you have the use the classical definition of n(A) / n(\omega)
which as the solution shows you do that by listing the possibilities in each set and dividing by the total number of possibilities(8)
compact epoch
Hmm i don't get the entries of the sets in this one
empty kindle
ok, let's list all the possibilities
I assume W is for the coat of arms, Z for number
empty kindle
(W, W, W)
(W, W, Z)
(W, Z, W)
(W, Z, Z)
(Z, W, W)
(Z, W, Z)
(Z, Z, W)
(Z, Z, Z)
those are all the possible combinations ^
event A says same 2 sides on last 2 throws
so we look at the list where the last 2 entries are the same
compact epoch
(W, W, W)
(W, W, Z)
(W, Z, W)
(W, Z, Z)
(Z, W, W)
(Z, W, Z)
(Z, Z, W)
(Z, Z, Z)
Oh that's more than 3!
empty kindle
(W, W, W)
(W, W, Z)
(W, Z, W)
(W, Z, Z)
(Z, W, W)
(Z, W, Z)
(Z, Z, W)
(Z, Z, Z)
there's 4
B says same sides on first and 3rd
(W, W, W)
(W, W, Z)
(W, Z, W)
(W, Z, Z)
(Z, W, W)
(Z, W, Z)
(Z, Z, W)
(Z, Z, Z)
compact epoch
Ohhhhh
empty kindle
now we want to check whether P(A and B) = P(A) * P(B)
compact epoch
i get it now LOL
empty kindle
so we need to see when does A and B happen together
compact epoch
Somehow my brain didn't register that W, W, W also counts as same side last 2 tries
empty kindle
(W, W, W) and (Z, Z, Z)
compact epoch
yep
empty kindle
so thats 2/8 for P(A and B)
P(A) = P(B) = 4/8
P(A) * P(B) = 1/2 * 1/2 = 1/4 = 2/8 = P(A and B), so yes they are independent
so that's checking pairwise, for complete independence you see whether P(A and B and C) = P(A) * P(B) * P(C)
compact epoch
Well at the end it says they are "pairwise independent" but not fully independent
empty kindle
so that's checking pairwise, for complete independence you see whether P(A and B...
.
essentially, knowing A does not affect the chances of B, but knowing A and C does very much affect the chances of B
compact epoch
Ah so that's what pairwise means?
empty kindle
yeah pairwise is you take 2 of them
you check all the possible pairs
compact epoch
Thanks dude you explained everything really well and fast 👍
empty kindle
np
cedar kilnBOT
cedar kilnBOT
Available help channel!
edgy sun
,,h(x)=\frac{-x^2+3}{4-x^2}
cedar kilnBOT
,,h(x)=\frac{-x^2+3}{4-x^2}
wraith daggerBOT
☠ cj Σ
edgy sun
i’m asked to decompose the composite function so that fog =h(x)
violet flume
what qualities have f and g
edgy sun
-x^2
edgy sun
what qualities have f and g
wait what are you asking?
edgy sun
yeah just one of them
violet flume
what about f and g is expected to be true
edgy sun
nothing it just has to be one of the ways to express it
violet flume
you should try
be as uncreative as possible
can you come up with one
try to cheat
edgy sun
try to cheat
That’s so unfair thooo
violet flume
i mean
edgy sun
i was thinking about it but i feel like a smart ass
violet flume
think of the easiest way
edgy sun
and i wanna know how else you can do it
edgy sun
mhm
violet flume
maybe a flip flop then
what if we flipped it
for g
then flopped it back
for f
edgy sun
it wouldn’t be the same right?
violet flume
why not
edgy sun
my first thought was g(x)=4-x^2 and f(x)=-x^2+3
violet flume
thats not composite
edgy sun
thats not composite
i know. It’s asking for two functions which when you do f of g you get
edgy sun
**☠ cj Σ**
this
but it was an issue since i forgot that i had to put the function in for both xs
violet flume
maybe you can try to think of a start and see if you can finish it
vivid wadi
take any p such that
p(p(x)) = x
f(x) = p(x)
g(x) = p(h(x))
then f(g(x)) = p(p(h(x)) = h(x)
violet flume
that is a fun way too 
edgy sun
maybe you can try to think of a start and see if you can finish it
i’m thinking something with making one thing a negative power and flipping all the xs into the denom
idk if that will work tho
violet flume
you can do some silly stuff
violet flume
are you talking about a flip flop?
vivid wadi
i’m thinking something with making one thing a negative power and flipping all t...
p(x) = x^(-1) mayhaps
violet flume
true
wraith daggerBOT
violet flume
hmm idk about that
vivid wadi
i’m thinking of something like this
,rotate
wraith daggerBOT
violet flume
what if we take g=x^2?
violet flume
i think its going to be hard to make this work
wraith daggerBOT
edgy sun
am i high or is that a correct answer
edgy sun
wait uhhh f(x)=4^-1+3 and g(x)=-x^-2-x^2
violet flume
wait uhhh f(x)=4^-1+3 and g(x)=-x^-2-x^2
its the same
edgy sun
why not try g=x^2
yeah that’s what i just thought about
wait why doesn’t this work
omg i hate math sometimes
it’s so cool but i find it so cool that i end up taking hours on my homework trying to make stuff more difficult
edgy sun
wait is that not right tho? can you not just switch the x^-2 and the 1/4
edgy sun
FUCK
violet flume
it doesnt work that way
edgy sun
when can you switch it?
violet flume
here, ill simplify, if you want
edgy sun
here, ill simplify, if you want
go for it
edgy sun
so f(x)=13/4?
wraith daggerBOT
jan Niku
edgy sun
why did i insist on doing it the weird way
violet flume
why did i insist on doing it the weird way
dont do it the weird way lol
i mean we can do crazy stuff here
but why
edgy sun
because i’m curious
vivid wadi
valid
violet flume
something that would work way better is maybe ...
.
sure
so i may try ...
try this, its more algebraic maybe
violet flume
make g some nice linear function
edgy sun
or denominator
violet flume
this is a fun puzzle
say g=x+3
now, when we square it, we get our x^2
but, extra junk!
not only do we get extra constants
but also extra x's
we can subtract x's, for sure! but, since its in composition, we settle for subtracting (x+3)s
does that make sense?
edgy sun
i think? wdym by subtracting (x+3)
violet flume
try this, as a easier problem
violet flume
but whats the problem?
edgy sun
extra constants and xs
violet flume
yea
wraith daggerBOT
jan Niku
violet flume
whats extra and whats needed?
edgy sun
we don’t need the 6x or the extra 4 added on
violet flume
right
wraith daggerBOT
jan Niku
violet flume
whats the problem
edgy sun
it’s g now
edgy sun
what would happen if i divided x^2+5 by x + 3
violet flume
dont divide
edgy sun
ok
violet flume
we can totally subtract x's
the thing is, when we do, we also have to subtract 3
at least if we wanna work with g's
edgy sun
WHAT IS HAPPENING
violet flume
but thats fine
violet flume
$x=(x+3) - 3 = g-3$
wraith daggerBOT
jan Niku
edgy sun
I prob should just do f(x)=(x+3)/(x+4) and g(x)=-x^2
violet flume
you are so close 😭
edgy sun
factor it?
violet flume
**jan Niku**
with our key here
edgy sun
g^2-x^2-9?
violet flume
no
violet flume
$h=g^2 -6\qty(g-3) -4$
wraith daggerBOT
jan Niku
violet flume
then $h= g^2 -6g +14$
wraith daggerBOT
jan Niku
violet flume
did i totally lose you?
edgy sun
**jan Niku**
here
violet flume
we create the highest order term
edgy sun
is where i got lost
violet flume
yes
edgy sun
ok
violet flume
we make x^2 with g, then fix the extra x crap we made
by writing x with g
then fix the constant crap we made
make sense?
violet flume
you just work down the list
this is what i would do, though
edgy sun
ok
violet flume
im not sure i follow
edgy sun
like just for the general equation
violet flume
easiest are identity and inverses
violet flume
no identity is
g=h
f=x
next is inverses
like g=1/h
f=1/x
inverses introduce domain problems
we can say we dont care though
edgy sun
ok
vivid wadi
yall still here
edgy sun
im just gonna say g(x)=h(x) then f(x)=x @violet flume
vivid wadi
valid
violet flume
im just gonna say g(x)=h(x) then f(x)=x <@331558346333224964>
this is the enlightened answer
come back to the larger puzzle again later
edgy sun
ok
violet flume
it will turn up often
edgy sun
ty
violet flume
dusky forge
help
cedar kilnBOT
@dusky forge Has your question been resolved?
velvet mortar
Be specific
cedar kilnBOT
@dusky forge Has your question been resolved?
cedar kilnBOT
cedar kilnBOT
Available help channel!
candid burrow
i thought that this is an alternating series where bn is decreasing on the interval (2,infinity) and lim as n approaches infinity bn = 0 so it converges but i keep getting it wrong. i dont understand
crimson sedge
i thought that this is an alternating series where bn is decreasing on the inter...
sounds good it should converge by alternating series test
what step r u having problems with
crimson sedge
try [1,inf)
candid burrow
cedar kilnBOT
@candid burrow Has your question been resolved?
candid burrow
Working on it
crimson sedge
i think u put (0.612372,inf)
candid burrow
i think the issue is my interval in which it is decreasing, it must be in interval form, not decimal
my friend said that i need to find the zeros from the derivative and use that for the bounds but 3x/8x^2-3 has no real roots or zeros
it doesnt accept (0, infinity) either
can you @ me when someone responds please?
cedar kilnBOT
@candid burrow Has your question been resolved?
plush wharf
How does n+1/5^n become 1/5 as limit goes inf
cedar kilnBOT
How does n+1/5^n become 1/5 as limit goes inf
lean lotus
substitution
vivid wadi
$\frac{n + 1}{5n} = \frac{1 + \frac{1}{n}}{5}$
wraith daggerBOT
kaue
