Recent Questions - Mathematics Stack Exchange |
- Finite sum $\sum_{i=2}^{5} \frac{i}{i-1}$
- Is this proof of s-plane to z-plane mapping correct?
- Does Magma have a labeling system for modular forms?
- Calculate p-val of new probability vs old probability
- Computation of row echelon form of matrix including trigonometric functions
- Is solution of $\frac{dy}{dx}=x^2+y^2 +1$ an odd function
- The Orbit Method and Finite Dimensional Unitary Representations of a Solvable Lie Group
- Intersection of the derived subgroup with a cyclic group
- Finding the probablity of a full house in poker with two wild cards
- How can I manipulate the way matrices multiply/divide to make their shapes match?
- Blocks diagonalization matrix
- Resolving the singularity arising from $\vec{x} \cdot \vec{x} = 0$
- What does it mean that a normal extension remain normal under lifting?
- Riemann-Roch theorem: a possible application
- Substitute variables in an expression in Sagemath
- Prove that $\circ$ defines a group action of $G$ on $L(X)$
- Solution to $((1-3i)^{8})^{\frac{1}{4}}$ in algebraic form.
- Prove that a function with $\cos(x)$ has zeros
- Test convergence of $\int_0^1x^p(\ln^q(\frac{1}{x}))dx$
- If $G$ is a group of order $p^n$, with $p$ prime and $n>1$. Prove that the order of every element of $G$ is a power of $p$
- Does the proof for primitive Pythagorean triples follow that the formula, $x=q^2-p^2, y=2pq, z=p^2+q^2,$ give all primitive Pythagorean triples?
- A parabola has focus F and vertex V, where VF=10. Let AB be a chord of length 100 that passes through F. Determine the area of triangle VAB.
- Drawing marbles until you run out of one color
- Connection between the cumulative distribution function and $p$-integrability
- Given k is a natural number. Find all functions $f: \mathbb{N} \to \mathbb{N} $ satisfy this
- Is there anyway to calculate $B_n$ from this equation?
- How to solve this recurrence with the iterative substituion method
- Let $f: [0 ,1] \to \mathbb{R} $ be continuous, prove $\lim_{n\to \infty} \int_0^1 f(x^n)dx = f(0)$
- Hyperbolic tangent integral $ \int_{-\infty} ^{\infty} e^{i x \frac h k} \tanh x \ dx$
- The Monty Hall problem
| Finite sum $\sum_{i=2}^{5} \frac{i}{i-1}$ Posted: 16 Feb 2022 11:55 AM PST I need to simplify that sum Finite sum $\sum_{i=2}^{5} \frac{i}{i-1}$ Can you help me, please? |
| Is this proof of s-plane to z-plane mapping correct? Posted: 16 Feb 2022 11:52 AM PST I am trying to prove that the left half-plane of s-plane is mapped into the interior of the unit circle in the z-plane. My proof is the following: We know that $s = σ + jΩ$ , $z = re^{jω}$ and $z = e^{sT}$ , so: $e^{sT} = re^{jω}$ $e^{(σ + jΩ)T} = re^{jω}$ $e^{σT}e^{jΩT} = re^{jω}$ Substituting $ω = ΩT$ (this is from DSP) we get $e^{σT}e^{jω} = re^{jω}$ and then $e^{σT} = r$ Since $T > 0$ it's obvious that when $σ < 0 => r < 1$. Is this correct? I am worried about the first three equations I used that I might unknowingly use as a fact what I am trying to prove. |
| Does Magma have a labeling system for modular forms? Posted: 16 Feb 2022 11:51 AM PST I am wondering if Magma has an internal labeling system for modular forms, or whether it is possible to access the LMFDB label for a modular form within Magma. I know that Magma has implemented such a system for elliptic curves, where we can access a curve using its Cremona label: for example, typing EllipticCurve("11a1"); in Magma returns the curve with Weierstrass model $y^2 + y = x^3 - x^2 - 10x - 20$. I am looking for something similar with modular forms. For example, by explicitly looking at Fourier coefficients, I can see that the newform f := Newform(CuspForms(Gamma0(3),6),1); in Magma is the modular form 3.6.a.a on LMFDB. Is there a way to access this label directly within Magma? |
| Calculate p-val of new probability vs old probability Posted: 16 Feb 2022 11:43 AM PST I have two probabilities, the old one is p1=0.72 and the new one is p2=0.84. I have to calculate the p-val of the benefit of the new probability. How do I calculate this? |
| Computation of row echelon form of matrix including trigonometric functions Posted: 16 Feb 2022 11:35 AM PST In our lectrue we consider the function $f:\{(r,\varphi,\theta)\in\mathbb{R}^3\mid r>0\} \to\mathbb{R}^3$, where \begin{align*} &f(r,\varphi,\theta):=\begin{pmatrix}r\cos(\varphi)\cos(\theta)\\r\sin(\varphi)\cos(\theta)\\r\sin(\theta)\end{pmatrix}\\ &Df=\begin{pmatrix}\cos(\varphi)\cos(\theta)&-r \sin(\varphi)\cos(\theta)&-r\cos(\varphi)\sin(\theta)\\\sin(\varphi)\cos(\theta)&r\cos(\varphi)\cos(\theta)&-r\sin(\varphi)\sin(\theta)\\\sin(\theta)&0&r\cos(\theta)\end{pmatrix} \end{align*} We must check if rank$(Df)=3$ by computing the row echelon form. Our solution says "If $\cos(\theta)=0$ then $Df=\begin{pmatrix}1&0&0\\0&0&1\\0&0&0\end{pmatrix}$ and If $\cos(\theta)\neq0$ then $Df=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$". Only if I assume $\cos(\theta)\neq0$ and $\sin(\varphi)\neq 0$, $\sin(\theta)\neq 0$, $\cos(\varphi)\neq0$, then after some tedious calculation I get $$Df=\begin{pmatrix}1&0&0\\0&\sin(\varphi)\cos(\varphi)\cos^2(\theta)&0\\0&0&\sin(\theta)\cos(\theta)\end{pmatrix}\implies Df=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$$ Only if I assume $\cos(\theta)=0$ and $\sin(\varphi)\neq 0$, $\cos(\varphi)\neq0$ I get $$ Df=\begin{pmatrix}1&0&0\\0&0&1\\0&0&0\end{pmatrix}. $$ I am not sure if further assumptions regarding $\sin(\varphi)$, $\sin(\theta)$ and $\cos(\varphi)$ are missing in the solution or if there is a trick to get rid of $\sin(\theta)$, $\sin(\varphi)$, $\sin(\theta)$ and $\cos(\varphi)$ without further assumptions? |
| Is solution of $\frac{dy}{dx}=x^2+y^2 +1$ an odd function Posted: 16 Feb 2022 11:47 AM PST I had this question in my exam which is of multiple select question (MSQ) type. I am sure about three options but confused about one. Clearly $\frac{dy}{dx}>0$, so $y$ is strictly increasing. Given that $y(0)=0$, we then have $y(x)<0$ for $x<0$ and $y(x)>0$ for $x>0$. Using this, options (B) and (D) get discarded and option (C) looks correct. But not getting any idea about option (A). Also I tried to solve this ODE but no method is working that I know. Any hint or help. Thanks. |
| The Orbit Method and Finite Dimensional Unitary Representations of a Solvable Lie Group Posted: 16 Feb 2022 11:33 AM PST Kirillov's Orbit Method, due to the work of Konstant, Auslander and Pukansky, works for solvable Lie groups. Is it possible, using this philosophy/method, to identify the finite dimensional unitary representations of a solvable Lie group in terms of properties of the associated coadjoint orbits? (Sorry for the crudeness of this question. I'm trying to evaluate whether or not to go down a rabbit hole, and the answer to this question will help me decide!) |
| Intersection of the derived subgroup with a cyclic group Posted: 16 Feb 2022 11:38 AM PST Let $G$ be a group and $G'$ be the derived subgroup. Let $C=\langle c \rangle $ be a cyclic subgroup of $G$. Is it true that the intersection $H=G'\cap C$ is trivial? I was thinking that if $g\in H$, then $g\in C$ and thus $g=c^{k}$, for some $k\in \mathbb{Z}$. Therefore, if $f$ is the homomorphism from $G$ to the abelianization $\bar{G}$ of $G$, then $f(g)=c^{k}$ and thus is not trivial. Hence since $\ker f=G'$ we have a contradiction. Is this idea correct or is there a case where $f(g)$ can be trivial in the abelianization? |
| Finding the probablity of a full house in poker with two wild cards Posted: 16 Feb 2022 11:28 AM PST Problem: Suppose that we have a standard poker deck and we add $2$ cards to it. These two cards added are jokers and are considered wild cards. What is the probability of getting a full house? Answer: |
| How can I manipulate the way matrices multiply/divide to make their shapes match? Posted: 16 Feb 2022 11:38 AM PST I'm trying to manipulate the way matrices multiply/divide in order to make sure that their shapes match while using the dot product. I'm not sure how well that was explained, so here is an example. Say I have this equation: $\frac{IX}{Y}$ and I know that the shapes are the following: $$I shape = (1000, 700)$$ $$X shape = (700,30)$$ $$Y shape = (30,10)$$ Notice : I will be using the dot product for multiplication and Moore-Penrose Inverse for division From this, we can notice that $I * X$ will work correctly with their shapes, and will result in the shape of $(1000, 30)$. If we go on, however, it would try to multiply $(1000, 30)$ and $(10, 30)$ because $\frac{1}{Y}$ will invert the shape. Because this doesn't work, I would change the equation to be $$(I * X) * Y^{-1}$$ where $Y^{-1}$ means that each element is inverted rather than the entire matrix. This works great for my purposes, however I will need to do this a lot with very large equations, so I want to create a program for it. To do that, I need a specific set of steps that I can follow to make the shapes match. EDIT: I just realized I forgot to put something essential in here. I want the shapes to match up, but I also want to it try to get a specific shape. In this case I would want it to go from $$\frac{(1000,700)*(700,30)}{(30,10)}$$ to $$(1000, 10)$$ |
| Posted: 16 Feb 2022 11:36 AM PST I have the following block matrix: $$\begin{pmatrix} A & B\\ B & A\end{pmatrix}$$ where $A$ and $B$ are $2 \times 2$ matrices. Is there a way to diagonalize it in blocks such that I get: $$\begin{pmatrix} C & 0\\0 & D\end{pmatrix}$$ where the matrices $C$ and $D$ (not necessarily diagonal) have some kind of relation with $A$ and $B$? Also, how would things change if I had instead: $$\begin{pmatrix} A & B\\ B & A^{\dagger}\end{pmatrix}$$ where $\dagger$ denotes the Hermitian conjugate? |
| Resolving the singularity arising from $\vec{x} \cdot \vec{x} = 0$ Posted: 16 Feb 2022 11:41 AM PST I'm just learning blowups and resolutions of singularities and I have been unable to find a clear and concise resource on how to resolve singularities in general. I understand that the concept of "resolutions of singularities" involves adding in a relation with respect to some projective coordinates. In this example, suppose that $\vec{x} = (x,y) \in \mathbb{R}^2$, then we introduce the projective coordinates $[X:Y] \in \mathbb{P}^1$, such that $xY = yX$, then WLOG assuming $X \neq 0$, we divide out by $X$, this leaves us with the relation $y = xY$, thus making this substitution, we find that: $x^2 + y^2 = 0 \iff x^2 + x^2Y^2 = 0 \iff x^2(1 + Y^2) = 0$. This does not seem to resolve my singularity, since I've still got a problem at $x = 0$, is anyone able to clarify for me what I am missing here? Most of the contructions that I've found just stop there, and it's not clear to me what I am doing, or what I am supposed to do. Thanks. |
| What does it mean that a normal extension remain normal under lifting? Posted: 16 Feb 2022 11:35 AM PST I consider the book "Algebra" by Serge Lang and on page 238 he has the theorem 3.4 saying that normal extensions remain normal under lifting. I don't see what he means by that, and therefore also his proof is not really clear to me. Can maybe someone explain this a bit more to me, because my TA told me something about a diagram that commutes and then the proof should be clear but I can't think about a diagram in this case. Thanks for your help. |
| Riemann-Roch theorem: a possible application Posted: 16 Feb 2022 11:47 AM PST Let be a finite set $S\subset \mathbb{C} \cup \{\infty\}$ and numbers $n(s)\in \mathbb{Z} \setminus \{0\}$, for each $s \in S$. In addition, also consider $$V = \{f:X\to \mathbb{C}; \ f \ \mbox{satisfies} \ (P_1), (P_2) \ \mbox{and} \ (P_3)\},$$ where: $P_1$) $f$ is holomorphic off $S$; $P_2$) If $s\in S$ has $n(s) > 0$, then $f$ has a pole of order at most $n(s)$; $P_3$) If $s\in S$ has $n(s) < 0$, then $f$ has zero order at least $|n(s)|$. Is it possible to determine the dimension of V? P.S: I saw some comments on this site that the Riemann-Roch Theorem can calculate or give an estimate for this problem, but I don't understand how it can be solved. I ask for help, since it is my first contact with Algebraic Geometry. |
| Substitute variables in an expression in Sagemath Posted: 16 Feb 2022 11:27 AM PST Somewhat similar to this question, I was trying to evaluate a Boolean expression given the right hand side variables in Sage. For simplicity, say, my Boolean expression is, $y=x_0+x_1$. For each of $(x_0,x_1) \in \{(0,0),(0,1),(1,0),(1,1)\}$, I want to evaluate $y$. This is the basic code block to get started. Note that, when I tried separate substitution, it works. But it does not work when I tried to automatically substitute variables. I tried multiple ways including various substitution,
Therefore, my query is how can I automatically substitute |
| Prove that $\circ$ defines a group action of $G$ on $L(X)$ Posted: 16 Feb 2022 11:49 AM PST Let $G$ be a group, $X$ a set, and define $(\cdot)$ an action of $G$ on $X$ as $(\cdot): G\times X \rightarrow X$ Let $L(X)=\{f:X\rightarrow \mathbb{C}\}$ If $g\in G$ and $f\in L(X)$ we define $\circ$ such that $\circ: G\times L(X) \rightarrow L(X)$ is defined as: $$(g\circ f)(x)=f(g^{-1}\cdot x)$$ My attempt Note that for $g=e$ we have: $$(e\circ f)(x)=f(e^{-1}\cdot x)=f(e\cdot x)=f(x)$$ Moreover, let $g_1\, \, g_2 \in G$, then $$g_1\circ(g_2\circ f)(x)= g_1\circ f(g_2^{-1}\cdot x)=f(g_1^{-1}\cdot (g_2^{-1}\cdot x))=f((g_1^{-1}\cdot g_2^{-1})\cdot x)=f((g_2\cdot g_1)^{-1}\cdot x)=(g_2\circ g_1)\circ (f)(x)$$ But here, I'm stuck, because I'm supposed to get: $$g_1\circ(g_2\circ f)(x)=(g_1\circ g_2)\circ (f)(x)$$ Can someone help me? Thanks. |
| Solution to $((1-3i)^{8})^{\frac{1}{4}}$ in algebraic form. Posted: 16 Feb 2022 11:53 AM PST I'm struggling with $$((1-3i)^{8})^{\frac{1}{4}}$$ I've tried raising it to the 8th power using De Moivre's Theorem and then finding it's 4th root but the angles are not solvable. Squaring $1-3i$ three times gives $16 (-527 + 336 i)$ which is't helpful either. I know I can express the answer with inverse trigonometric functions but I can't find a way to get to $2(4+3i)$. |
| Prove that a function with $\cos(x)$ has zeros Posted: 16 Feb 2022 11:39 AM PST The ProblemI'm trying to find the zeros of this function in the $[0, +\infty]$ interval: $$f(x) = e^{-x^2} - \cos(x)$$ What I've triedI know that $cos(x)$ is a periodic function limited by $[-1, 1]$ and that $e^{-x^2}$ asymptotically approaches $0$ as $x \rightarrow +\infty$, therefore, both $\cos(x)$ and $e^{-x^2}$ have equal values in many points in the $[0, +\infty]$ interval . But i don't know how to manipulate the function formula in order to find the infinite amount of zeros of this in function in the $[0, +\infty]$. Any ideas on how can i do that? I've thought about reviewing some materials related to numerical analysis ( like the method of successive approximations ) but found nothing that could help me. |
| Test convergence of $\int_0^1x^p(\ln^q(\frac{1}{x}))dx$ Posted: 16 Feb 2022 11:28 AM PST Test convergence of $\int_0^1x^p(ln^q(\frac{1}{x}))dx$ My work. Let $ln(x^{-1})=y$ then integral becomes $\int_\infty^0-e^{-yp-y}y^qdy$ = $\int_0^\infty e^{-yp-y}y^qdy$=$\int_0^1 e^{-yp-y}y^qdy$ + $\int_1^\infty e^{-yp-y}y^qdy$ How continue ?I want to bring these in the form $\int_0^1\frac{1}{x^a}$ and $\int_1^\infty\frac{1}{x^a}$ where I know where integral is convergent.I saw similar questions but I couldn't understand. I want to solve this question without using Gamma function I am getting $(\frac{1}{p+1})^{q+1}\int_0^\infty z^qe^{-z}dz $ which is convergent when $q>−1$ but can't get values of $p$ from here. |
| Posted: 16 Feb 2022 11:37 AM PST Here is my approach: If $a\in G$, let's consider a subgroup $H=[a]$. Lagrange's theorem tell us that $o(H) \mid p^n$. But since $p$ is prime, the order $H$ must be the power $p\leq n$, with $k\leq n$. In particular, $o(a)=p^k$ |
| Posted: 16 Feb 2022 11:32 AM PST Essentially, I am aware of the proof that if a primitive Pythagorean triple is of the form $x^2+y^2=z^2,$ then $x=q^2-p^2,$ $y=2pq$ and $z=p^2+q^2.$ But I'm not going to write this proof here. My question was, does it follow that for any $p,q$ element of the integers, that one can follow this formula and find a primitive Pythagorean triple? Or is it simply the case that, any primitive Pythagorean triple will have this form? This is not homework related, so feel free to give me in depth proofs/explanations. |
| Posted: 16 Feb 2022 11:41 AM PST
This is an olympiad question which I came across last week. I really don't have any idea where to start. I think the information provided in the question is not even enough to solve the problem. I only know that for a parabola $y^2 = 4ax$, the length of the focal chord through $t$ is given by $a\left(t+\dfrac1t\right)^2$. Can anyone check the problem if it's correct? If yes, then how may I proceed? Any hint would be enough. |
| Drawing marbles until you run out of one color Posted: 16 Feb 2022 11:53 AM PST You have a bag with marbles and draw without replacement. The marbles have $k$ distinct colors and there are exactly $n$ marbles of each color. All marbles are equally likely to be drawn. You continue drawing until the bag runs out of a color so you have drawn all $n$ marbles of that color. What is the expected number of marbles you need to draw and what is the probability distribution to model this process? You clearly have do draw at least $n$ marbles and at most $k(n-1) +1$ by the pigeon hole principle. This looks similar to the negative hypergeometric distribution but it doesn't quite fit because I want to stop drawing the moment I run out of any color not just one particular color. Simplest example: There are $2$ red and $2$ green marbles in the bag. Without loss of generality let the first marble drawn be red. There is now a $1/3$ probability that the second marble is also red and I stop. If the second marble is green, the third marble will be either red or green and the bag will be out of that color so I stop no matter what. So with probability $2/3$ I will draw $3$ marbles. Hence the expected number of marbles drawn is $2\cdot (1/3) + 3\cdot (2/3)=2.67$. |
| Connection between the cumulative distribution function and $p$-integrability Posted: 16 Feb 2022 11:40 AM PST Let $p \geq 1$ and $l > 0$. Suppose that $X$ is a non-negative random variable with a CDF satisfying for some constant $c > 0$, $\lim_{x\to+\infty}(x^l(1 - F_X(x))) = c$. I have to show that a non-negative random variable $X$ is $p$-integrable iff $p < l$. I already know that the expected value of a non-negative r.v. can be expressed as $E(X) = \int_0^\infty(1 - F_X(x))dx$, and for $X^p$, $E(X^p) = \int_0^\infty px^{p-1}(1 - F_X(x))dx$. I'm not looking for a complete answer, but some hints would be nice as currently I don't have any clue on how to start the proof. To be specific, I don't know how to use the fact that $\lim_{x\to+\infty}(x^l(1 - F_X(x))) = c$ with the CDF form of the moment $p$ of $X$. |
| Given k is a natural number. Find all functions $f: \mathbb{N} \to \mathbb{N} $ satisfy this Posted: 16 Feb 2022 11:26 AM PST This function need to satisfy $f(m)+f(n)$ is divisor of $(m+n)^k,$ with $m,n$ are natural numbers. In this problem, $\mathbb N$ does not include $0.$ In the case k=1, I managed to solve this by replacing $m$ and $n$ with 1;p-1;p (p is prime number) and received $f(n)=n.$ For other $k,$ I have no idea. |
| Is there anyway to calculate $B_n$ from this equation? Posted: 16 Feb 2022 11:56 AM PST $1-y=\sum_{n=1}^\infty B_n (e^{n\pi}-e^{-n\pi})\cos(n\pi y)$ I encountered this equation when solving a Laplace equation and I believe I didn't solve the $B_n$ correctly. The way I did it was the usual Fourier's trick: $B_n(e^{n\pi}-e^{-n\pi})=2\int_0^1(1-y)\cos(n\pi \ y)dy=\frac{2(1-(-1)^n)}{n^2\pi^2}$. However, after plotting the result, I found the solution doesn't match one of the boundary conditions. Then I realized Fourier's trick might be invalid here because the term $e^{n\pi}-e^{-n\pi}$ depends on $n$ so I can't just take them out of the integral with $B_n$. I wrote this program to plot the results.
After realizing that, I found myself stuck so I wonder if anyone knows how to calculate $B_n$ in this equation. Thanks a lot! The Laplace equation and what I've got are presented here: |
| How to solve this recurrence with the iterative substituion method Posted: 16 Feb 2022 11:34 AM PST Note: I know we can solve this by other method ( Master Theorem) but i need to solve this by iterative substitution which means i need to show the pattern for i =1 i=2 i=3 and then give a general formula So there's something i am not doing right Could anyone help ?Thanks |
| Let $f: [0 ,1] \to \mathbb{R} $ be continuous, prove $\lim_{n\to \infty} \int_0^1 f(x^n)dx = f(0)$ Posted: 16 Feb 2022 11:39 AM PST Let $f: [0,1] \to \mathbb{R}$ be continuous, prove $\lim_{n\to \infty} \int_0^1 f(x^n)dx = f(0)$ This makes some sense looking at it. I have only the Regulated Integral definition of integration to work with :https://en.wikipedia.org/wiki/Regulated_integral It uses sequences of step functions with a partition over a closed interval. So I need to solve the integral prior to taking the limit but not sure how to really get what I need. Since $f $ is continuous function there exists a sequence $(\varphi_n)_{n \in \mathbb{N}}$ of step functions such that $\lim_{n \to \infty} \sup_{x \in [0,1]} \mid f(x) - \varphi_n(x)\mid \,= 0$. Moreover, $\int_0^1 f(x)dx := \lim_{m \to \infty} \int_0^1 \varphi_m(x)dx $ so it should be the case that $\int_0^1 f(x^n)dx := \lim_{m \to \infty} \int_0^1 \varphi_m(x^n)dx $ so I assume that it would be that $\lim_{n \to \infty } \int_0^1 f(x^n)dx := \lim_{n \to\infty} (\lim_{n \to \infty} \int_0^1 \varphi_m(x^n)dx )$ Now, $ \int_a^b \varphi(\eta)d\eta := \sum_{j=0}^{N}\varphi(\eta_j)(\sigma_{j+1}-\sigma_j)$ where $(\sigma_j)_{j=0}^{N+1})$ is a partition of $[a,b]$ and $\eta_{j} \in (\sigma_j,\sigma_{j+1})$ such that each block of the partition is constant so choice of $\eta_j$ is not particularly important. So should have $ lim_{n \to \infty}(lim_{m \to \infty}\int_a^b \varphi_m(\eta^n)d\eta := lim_{n \to \infty}(\lim_{m \to \infty}\sum_{j=0}^{N}\varphi_m(\eta_j^n)(\sigma_{j+1}-\sigma_j))$. This is where I get stuck. |
| Hyperbolic tangent integral $ \int_{-\infty} ^{\infty} e^{i x \frac h k} \tanh x \ dx$ Posted: 16 Feb 2022 11:50 AM PST Find the integral of $$ \int \limits _{-\infty} ^{\infty} e^{\Bbb i x \frac h k} \tanh x \ \Bbb dx$$ I tried to expand this but that didn't help. |
| Posted: 16 Feb 2022 11:55 AM PST I was watching the movie $21$ yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability. The question goes a little something like this (I'm paraphrasing, but the numbers are all exact): You're on a game show, and you're given three doors. Behind one of the doors is a brand new car, behind the other two are donkeys. With each door you have a $1/3$ chance of winning. Which door would you pick? The character picks A, as the odds are all equally in his favor. The teacher then opens door C, revealing a donkey to be behind there, and asks him if he would like to change his choice. At this point he also explains that most people change their choices out of fear; paranoia; emotion and such. The character does change his answer to B, but because (according to the movie), the odds are now in favor of door B with a $1/3$ chance of winning if door A is picked and $2/3$ if door B is picked. What I don't understand is how removing the final door increases the odds of winning if door B is picked only. Surely the split should be 50/50 now, as removal of the final door tells you nothing about the first two? I assume that I'm wrong; as I'd really like to think that they wouldn't make a movie that's so mathematically incorrect, but I just can't seem to understand why this is the case. So, if anyone could tell me whether I'm right; or if not explain why, I would be extremely grateful. |
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