What is the probability of rolling two 1's on a pair of dice, if you get to roll 24 times?

so, it's a 1 in 36 shot... but it won't come out to 2/3rds the time... and i'm confused as to why

35/36 times you will fail at rolling two ones. If you do this 24 times, the probability of failing to roll two ones is

(35/36)^24 = .508

Thus, the probability of rolling at least two ones once in 24 tries is 49.2%.

Okay, but now applying more snake eyes probability...

Suppose that we have two identical looking urns, call them #1 and #2. Urn #1 has eight red balls and two black balls in it; urn #2 has two red balls and eight black balls. I’m going to select one of the urns at random by rolling two six-sided dice, and picking urn #1 just in case the total on the two dice is two (i.e., I rolled a double-ace). I won’t tell you which urn I picked. You then draw a ball out of the mystery urn. If it is a red ball, how likely is it that you drew from urn #1?

Good luck, I can NOT get this answer for the damned life of me.

aparently I can magically use this
P(p | q) = P(p) x P(q | p) / P(q)
to help me answer

Info:
Urn #1 has P(0.8) of drawing a red ball and Urn #2 has a P(0.2) of drawing red ball.
A roll is made on 2d6 and when snake eyes is rolled, agent #1 selects urn 1, else agent #1 selects urn 2.
Agent #2 draws a ball that is red, how likely is it that agent #2 drew from urn 1?

Essentially, 1/36 times we get an 80% chance to draw a red ball and 35/36 times we get a 20% chance to draw a red ball.

What is the chance that a red ball is drawn if we don’t know what urn we’re selecting from?
0.022222 + 0.1944444 = 0.2166666

P(p | q) = P(p) x P(q | p) / P(q)

So…

P(drew from urn #1 | red ball) = P(1/36) x P(0.216666 x 1/36) / P(0.216666)

is where i'm at so far, but it doesn't look correct

0.0007712 seems ridic small

You're on the right track.

P(#1 urn) = 1/36
P(red|#1 urn) = 4/5
P(red) = P(red |#1 urn)*P(#1 urn) + P(red|#2 urn)*P(#1 urn) = (4/5)(1/36) + (1/5)(35/36)

I'm too lazy to calculate the fraction, but I think it should be around 10-11%

How would you go about solving this word problem:

An apparel shop sells skirts for $45 and blouses for $35. It's entire stock is worth $51,750. But sales are slow and only half the skirts and two-thirds the blouses are sold, for a total of $30,600. How many skirts and blouses are left in the store??

My guess: 45x+35y= 51,750
.5x+.33y= 21,150 and just solve these system of equations?? Any help would be appreciated.

Wasn't too sure where to post this but figured I'd get some decent responses here.

Currently at my final year of university and we're expected to deliver a final year project based on a specific topic related to mathematics. I've chosen the Application of Mathematics in Business and Finance. I've got some material for the first two chapters of my project but struggling to think of something for my final two.

I'm looking for an interesting event that happened in the world of finance/economics that I can use mathematical concepts to explain. Like the Wall Street Crash or something.

Or even dipping in and out of popular culture.

Anyone with any thoughts/ideas?

So number of shirts*45$ + number of blouses*35$=51750$? Makes sense.
But half the shirts + 2/3 of the blouses=21150$ doesn't. You forgot the prices.

Is it possible to find a function g(x) ∈ L¹([0,1]) so that:

∫_[0,1]xⁿg(x)dx = δ_n1

For n = 0, 1, 2,..., N where N is finite?

Yes.

Prove that the map

is a bijection between the polynomials of degree less than N and R^(N+1).

There are tons of interesting events that have happened in finance. Of course, interesting in the finance world usually means that someone lost a lot of money. Here are a couple of subjects, you can Google if you want to know more:

The recent credit crisis.
Tech market/dotcom bubble.
Fall of LTCM.
Japan housing/asset bubble.
Russian Ruble crisis.
Mexican Peso crisis.

Furthermore, there are individual investors who may be interesting, eg. Buffet and Soros.

You mention that it is a math assignment. I've always been curious to what degree the market is efficient. Is it beatable? Is it only beatable for a select few? It has been done before, but since there are a lot of data of financial transactions, this subject should lend itself well to a math paper.

Is "final year of university" the final year of a undergraduate or postgraduate degree? There are also tons of nice math in mathematical finance - this could maybe be interesting for you to look into. Depending on your math level I could point you towards some of it, if you're interested in stuff like that.

checktheriver -

Thanks for the reply. Sadly I do not understand your hint, but I have come upon a simpler way of showing the existence of such a g(x). Assume g(x) is in the space spanned by {1,x,...,x^N}. Then we have



So that



Now our conditions for l=0,1,...,N imply the system of N+1 equations



And the coefficient matrix here is a Hilbert matrix, hence invertible. Thus we can solve for the coefficients of g(x).

But this is not actually my homework problem. My homework is this: let P_N be the space of real polynomials of degree ≤ N, taken as a subspace of L^∞([0,1]). Define a bounded linear functional δ on P_N by:



Now by the Hahn-Banach theorem, δ extends to a bounded linear functional Δ on all of L^∞([0,1]) with the same norm. Question: is it possible to find a fixed g(x) ∈ L¹([0,1]) so that it holds...



...for all f(x) ∈ L^∞([0,1])?

Ok, so evidently the g(x) we found above, which satisfies the condition ∫_[0,1]x^kg(x)dx = δ_k1, yields a bounded linear functional Φ_g on L^∞([0,1]) so that:



But...does this tell us anything about Δ? (God, fml I'm incompetent.)

No that was exactly what I was saying

btw to show that the linear map is invertible (=bijective) you don't need to make calculations (or refer to Wikipedia)...

Indeed if for some P of degree less than N,

for k=0,..N, then
,
hence P^2=0 on [0,1], and P having infinitely many zeroes is the 0 polynomial.
So the linear map is injective, and since both spaces have same dimension (N+1), it is bijective.

--
As to your other question, I don't know...I might be missing something as I'm not an expert in functional analysis, but I'm not sure you can really say anything the way the question is worded. If the question is "Is there a \Delta pronlonging \delta, of the form \Phi_g for some g" then you have shown that it is true, but I'm not sure if you can say anything about any \Delta you get from Hahn-Banach.

Oh, now I see the point---thanks! If I figure out any more I will let you know.

I have a really basic linear algebra question, we just started learning about the concept of vector spaces and subspaces so this is probably really trivial to anyone else.

Which matrices are "vectors" contained in the smallest subspace containing square matrix A where row1 is 2, -2 and row2 is also 2, -2?

I was thinking a subspace where all matrices are 2x2 and row1=row2 and column1= -column2.

But how do I know or check to see if this is the smallest subspace that A exists in?

Let's abstract away the fact that they are matrices.

What you're asking is: What vectors are contained in the smallest subspace containing the vector v?

Well, subspaces are closed under linear combinations, so we start by taking all linear combinations of the vectors we know are in there, namely v. Those are just constant multiples of v. We can check that this is in fact a subspace (I'll leave that to you).

Now applying this to the matrix you asked about...
The subspace is all constant multiples of M, so all matrices of the form:
x -x
x -x

This is a one-dimensional subspace, as expected.

Thanks, that makes sense. Thats basically how my reasoning went but I am new to this stuff so I am not very confident yet. (Also your form for the matrix looks a lot more "mathy" than my written out sentence )

Hi guys, have a quick question. It doesn't belong in this category per se, but didn't think it deserved its own thread.

So, I'm studying a Master in Mathematical Finance. I am contemplating pursuing a PhD in applied math (probability theory/stochastic simulation). I am reasonably satisfied with my mathematical skills - they are sufficient to solve most exercises/exams - but I really think that I have a problem when I have to solve things myself, ie. solve difficult exercises where I don't have a hint from my lecturer. Oftentimes the exercises we solve are neatly fit in the stuff we learn in the lectures which makes it easy enough since you know what kind of knowledge you should apply.

My problem comes when I study on my own and in general I feel like my (mathematical) analysis-skills are sub-par, since I've never done a degree in pure math. Up until know, this haven't gotten me into too much trouble, but if I'm going to apply for a phd I figure I probably should get better at stuff like this.

Do you guys have any tips and/or books I could consult?

Standard. Math is hard. If you enjoy learning/doing math, go for it. If you're going for a PhD because you think it will improve your job prospects, don't go; it will do the opposite in general. GL.

Thanks for the encouraging words.
It is not for the job prospects and certainly not for money, but more, like you say, that I enjoy learning and would like to dig deeper into this stuff.

I feel like I am "bright enough" to do a phd, but when I talk with current phd-students they seem to know everything and they are very adept at solving even difficult exercises/problems and can answer most questions at their feet. Of course this is a product of them doing research/studying for a living but I also feel like their basic mathematical understanding is deeper than mine and I would be uncomfortable teaching without this background knowledge they seem to have.

Spend enough time doing this stuff day in & day out, and spend enough time in the company of these really smart people, and you'll start to mimic them and learn from them. It's like how people's accents can change after some time if they make a regional move. Also, you live in Alabama long enough, you'll use the word y'all sometimes.

What's considered general knowledge for grad students varies greatly university to university. Go someplace you're comfortable. In other words, Harvard math may not be for you, but that's ok. The admissions process does a reasonable job of putting students in the right place anyway.

My experience: I was at Michigan. We had ~125 grad students. Backgrounds/work ethics/interests/talents varied greatly. Everyone just found his/her niche, and we all drank a lot of beer.

Physics would also be a reasonable route for applied math. You get to learn the dirty tricks that would make Wyman squeal.

Also if you want to go the quant route, MFE or equivalent then PhD in any hard science with some computation should make you very competitive. Physics, pchem, math, CS, etc.

I also have a related question: Due to poker I don't really need a funded phd. Does this ease the application process? Ie. is it easier to get a phd at a good university?

In theory. However, you probably don't want to go somewhere that doesn't admit you with funding. Here's why: getting a PhD is not something where you can just go thru the motions and end up graduating. It's really hard, and it requires a lot of work -- beyond coursework. If you are far enough in skill behind everyone else you're going to school with, you'll struggle in courses and probably in finding an advisor that's any good. And the level of research expected of graduates there might not be where you're at. Not to mention, when you come out of grad school, you'll need a good letter of recommendation (or several).

Damn near everyone in math grad school gets funding. Usually, this is by teaching calculus (it's part of the experience; suck it up and do it. Maybe you'll like it). My advice: go somewhere you get in legit. If there's a place you have your heart set on and you are waitlisted, you can consider trying to negotiate with them that you'll pay the first year or something. Paying for 5 years of grad school you can get for free just makes no sense, especially if it dumps you in a situation where you're a fish out of water.

Just my $0.02.

edit: and i dont know how much you play, but i wouldn't plan on playing more than like 5-10 hours a week or so if you decide to go to grad school. If you want a PhD, you should really go at this 100%. Part time PhDs just dont work well.