https://math.codidact.com/categories/41/feedNew Posts - Q&A - MathematicsMathematics - Codidact2021-06-10T23:13:26Zhttps://math.codidact.com/posts/282061How does Pr(an event next year) $= 1/100$ imply Pr(at least one of these events occurring in the next 25 years) $> 1/5$?Chgg Clouhttps://math.codidact.com/users/535642021-06-06T23:45:21Z2021-06-10T23:13:26Z<p>Please see the embolden phrase below. It appears to equalize, or at least relate, Pr(an event next year) $= 1/100$, with Pr(at least one of these events occurring in the next 25 years) $> 1/5...https://math.codidact.com/posts/282055How's the outcome of airplane crashes almost always certain, but the probability is not? Chgg Clouhttps://math.codidact.com/users/535642021-06-06T08:26:15Z2021-06-07T07:49:58Z<p>Please see the bolden phrase and Table 12.1 below. For plane crashes, did the author mix up which (probability or outcome) is ambiguous, and which is precise?</p>
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<li>Isn't the outcome of...https://math.codidact.com/posts/281987What story and two-digit Natural Numbers best fit Bayes' Theorem chart?DNBhttps://math.codidact.com/users/536282021-06-02T07:36:40Z2021-06-06T23:55:36Z<p>To complete the table below most comfortably for teenagers,</p>
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<p>what are the simplest stories?</p>
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<p>what natural numbers $\le 99$ contrast the base rate fallacy t...https://math.codidact.com/posts/282060 Why isn't $\Pr(diseased|+ test) = \dfrac{\text{number of true positives}}{\text{number of false positives}}$? Chgg Clouhttps://math.codidact.com/users/535642021-06-06T23:40:32Z2021-06-06T23:41:45Z<ol>
<li>Technically, the $\color{red}{10}$ quoted below should be 9.99, because $1% \times 999 = 9.99%$. Anyways, why $\Pr(diseased|+ test) = \dfrac{1 \text{ true positive}}{9.9 \text{ false posi...https://math.codidact.com/posts/282049Why paradoxical to accept \$3, and avoid betting on Dushanbe's temperature for a prize of \$10?Chgg Clouhttps://math.codidact.com/users/535642021-06-06T06:58:34Z2021-06-06T23:07:58Z<p>If you don't know Dushanbe's past or historical temperatures, then isn't it rational to accept the $3 for sure? It doesn't feel rational to bet whether the temperature's low or high in Dushanbe,...https://math.codidact.com/posts/282056What does "individuals discount the future in a constant manner" mean?Chgg Clouhttps://math.codidact.com/users/535642021-06-06T08:30:00Z2021-06-06T08:30:00Z<ol>
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<p>Is the embold phrase referring to Present Value?</p>
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<p>But how does Present Value relate to the example below with the "online bookstore"?</p>
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...https://math.codidact.com/posts/282052If money's less valuable in the two-bullet case of the Russian Roulette problem, then ought you pay more to remove a bullet when the gun has $\ge 2$ bullets? Chgg Clouhttps://math.codidact.com/users/535642021-06-06T07:08:55Z2021-06-06T07:08:55Z<p>The emboldened sentences feel contradictory. On one hand, "they are equal reductions in the probability of death". On the other, "money is less valuable in the two-bullet case sin...https://math.codidact.com/posts/282051Why should the near-zero prior probability, of 9/11/2001 attacks on the World Trade Centre, overwhelm any subsequent response that may have lowered the objective risk?Chgg Clouhttps://math.codidact.com/users/535642021-06-06T07:03:16Z2021-06-06T07:04:52Z<p>Can you please expound the embold phrase below? Can this be calculated?</p>
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<p> It is notoriously difficult, however, to assess what one’s prior assessment of the risk would ...https://math.codidact.com/posts/282046Why rational to be indifferent between two urns, when urn A has 50-50 red and white balls, but you don't know urn B's ratio? Chgg Clouhttps://math.codidact.com/users/535642021-06-05T23:38:20Z2021-06-06T00:12:43Z<p>Please see the embolden sentence below. Assume that I'm risk adverse and "prefer the known chance over the unknown". Why's it irrational for me to choose A?</p>
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<p> ...https://math.codidact.com/posts/282047Why's the unique sub-game perfect equilibrium, that the first player should offer around $1.25 to player 2?Chgg Clouhttps://math.codidact.com/users/535642021-06-05T23:53:46Z2021-06-05T23:53:46Z<p>Please see the bolden phrase below. Let's abbreviate player $j$ to $Pj$. Even if this is the unique sub-game perfect equilibrium, it feels unnecessarily risky and irrational to me. Why wouldn't ...https://math.codidact.com/posts/281010De-mystifying tricks – If $\{x_n\}$ converges, then Cesaro Mean converges.VLDRhttps://math.codidact.com/users/536962021-03-09T06:12:46Z2021-06-05T16:19:25Z<blockquote>
<p><strong>Exercise 2.3.11 (Cesaro Means).</strong> (a) Show if ${x_n}$ is a convergent sequence, then the sequences given by the averages ${\dfrac{x_1 + x_2 + ... + x_n}{n}}$ converg...https://math.codidact.com/posts/280653How can I visualize $\lim\limits_{x \rightarrow \pm \infty} f(x) = \lim\limits_{t \rightarrow 0^{\pm}} f(1/t)$?VLDRhttps://math.codidact.com/users/536962021-02-02T21:22:23Z2021-06-05T15:38:14Z<p>I'm not asking about <a href="https://math.stackexchange.com/a/3983012/872474">the proof</a> that I already understand. I'm longing to understand this graphically. As you can see, I added $1/t$...https://math.codidact.com/posts/282014How to quantify "married people are less likely than the average person to smoke", "smokers are less likely than the average person to be married"?Chgg Clouhttps://math.codidact.com/users/535642021-06-04T04:43:32Z2021-06-05T14:49:27Z<p>When I first saw inequalities (1) and (2) below, I quantified them as:</p>
<p>$\color{red}{\text{1.1. married people < all smokers}}$</p>
<p>$\color{red}{\text{2.1. and married smokers <...https://math.codidact.com/posts/282031Why are you permitted to define $1 − 1 + 1 − 1 + . . .$? Chgg Clouhttps://math.codidact.com/users/535642021-06-05T07:50:25Z2021-06-05T07:54:43Z<p>Please see the embold phrase below. Why doesn't $1 − 1 + 1 − 1 + . . .$ possess an intrinsic, Platonic objective meaning? The best way to showcase my confusion, is to burlesque the Riemann Hyp...https://math.codidact.com/posts/282020What was Justice Scalia's mathematical mistake in Penry v. Lynaugh (1989)?Chgg Clouhttps://math.codidact.com/users/535642021-06-04T07:51:58Z2021-06-05T02:05:16Z<p>Please see the bolden phrase below. Please don't hesitate to reduce the amount of quotation, which I know is lengthy, whilst preserving enough context.</p>
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<p> But even grant...https://math.codidact.com/posts/282016Why mustn't the proportion of smokers among married people be the same as the proportion of smokers in the whole population?Chgg Clouhttps://math.codidact.com/users/535642021-06-04T04:49:56Z2021-06-04T16:50:39Z<p>Please see the embolded phrase below. When I read this for the first time, I didn't see this problem at all, and this problem didn't present itself immediately to me. After rereading this four t...https://math.codidact.com/posts/282015How to intuit : married smokers × all people < all smokers × all married people ? Chgg Clouhttps://math.codidact.com/users/535642021-06-04T04:46:23Z2021-06-04T04:46:23Z<p>How can I intuit inequality (3) <a href="https://math.codidact.com/posts/282014">in my previous post</a>? The author intuits inequalities (1) and (2), but not (3).</p>
<p>Scilicet, how can you...https://math.codidact.com/posts/282013Why do 3D mental pictures usually suffice for high-dimensional geometry?Chgg Clouhttps://math.codidact.com/users/535642021-06-04T04:32:18Z2021-06-04T04:32:32Z<p>Kindly see the embolded phrases below. The author doesn't expound why the 3D "mental pictures" are "usually enough". Scilicet, why doesn't "this impoverished vision"...https://math.codidact.com/posts/282012How can you foretell if a problem is one whose solution admits a simple mathematical description?Chgg Clouhttps://math.codidact.com/users/535642021-06-04T04:31:47Z2021-06-04T04:31:47Z<p>Kindly see the emboldened phrase below. What kind of problems is the author referring to?</p>
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<h3>THE UNREASONABLE EFFECTIVENESS OF CLASSICAL GEOMETRY</h3>
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<blockq...https://math.codidact.com/posts/281985Why can an easily graphable definite integral, be labyrinthine to evaluate?VLDRhttps://math.codidact.com/users/536962021-06-02T06:29:00Z2021-06-02T06:52:58Z<p>How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to symbolically integrate definite integrals, when their graph looks so unremarkable and straightf...https://math.codidact.com/posts/281955Notation for Euclidean NormGuilherme Costahttps://math.codidact.com/users/534532021-05-30T17:45:02Z2021-05-30T19:26:35Z<p>What is the most common notation for the Euclidean Norm? $||x||_{2}$ or $||x||$ ?</p>
<!-- g: js, mdit -->https://math.codidact.com/posts/281864How to show if a set is simply connected?kreijstalhttps://math.codidact.com/users/540012021-05-23T20:44:27Z2021-05-23T22:22:28Z<p>How do I show that the following set is not simply connected?</p>
<p>$M= \mathbb{R}^3\setminus\{(x,y,z)\in \mathbb{R}^3|y=z=1\} $</p>
<p>I am aware that if a set has a hole, then it isn't simp...https://math.codidact.com/posts/281724proving relative lengths on a secantmsh210https://math.codidact.com/users/81062021-05-13T10:26:34Z2021-05-14T14:07:28Z<p>My kid was given this question. (Well, my statement of it actually includes some results that my kid had to find in previous parts of the question.)</p>
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<p>Triangle $ABC$ is equil...https://math.codidact.com/posts/281630difference between quotient rule and product ruleIstiakhttps://math.codidact.com/users/539222021-05-08T13:42:52Z2021-05-13T10:20:45Z<p>Product rule :</p>
<p>$$\frac{d}{dx} f(x)g(x)=f'(x)g(x)+f(x)g' (x)$$</p>
<p>Quotient rule :</p>
<p>$$\frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}$$</p>
<p>Suppose, the...https://math.codidact.com/posts/281625Why is -1 needed to get 65,535 from a 2^16 binary power calculation?JohnDoeahttps://math.codidact.com/users/363632021-05-08T08:42:58Z2021-05-08T09:19:03Z<p>It was stated in the Wikipedia article <a href="https://en.wikipedia.org/wiki/65,535">65,535</a>:</p>
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<p>65535 occurs frequently in the field of computing because it is <code>2^16...https://math.codidact.com/posts/281488BODMAS and PEMDASIstiakhttps://math.codidact.com/users/539222021-04-24T05:15:17Z2021-04-24T05:15:42Z<p>There's very known two rules BODMAS and PEMDAS. Lot more rules exist like GEMDAS, GIDMAS etc. But, BODMAS and PEMDAS are very known.</p>
<h4>BODMAS : </h4>
<p><code>B : Bracket</code><br/>
<c...https://math.codidact.com/posts/280168How can I deduce which operation removes redundacies?DNBhttps://math.codidact.com/users/536282020-12-30T03:11:56Z2021-04-09T12:17:42Z<p>Please don't answer by working backwards from the answer, or by appealing to arithmetic. Act as if you're learning this for the first time.</p>
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<p>How can I deduce which operation ...https://math.codidact.com/posts/280799Is this topology basis dependent?celtschkhttps://math.codidact.com/users/80562021-02-14T19:32:06Z2021-03-30T18:08:26Z<p>Consider a topological field $K$ and an algebraic(!) vector space $V$ over $K$, that is, $V$ has not (yet) a topology defined on it. I'm particularly interested in the case where $V$ has infinit...https://math.codidact.com/posts/281005Intuitively, why's X% of Y = Y% of X?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:09:37Z2021-03-09T09:27:03Z<p>I didn't spot this trick until I read <a href="https://redd.it/7yvxw4">this</a>:</p>
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<p>LPT: X percent of Y is equal to Y percent of X.</p>
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<p>I know that multipli...https://math.codidact.com/posts/281006In If and Only If Proofs, why's the proof for one direction easier than the other?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:27:22Z2021-03-09T09:20:44Z<p><a href="https://math.stackexchange.com/q/3069488">This list</a>. <a href="https://math.stackexchange.com/q/3069488">this question</a> substantiates proofs where one direction can be proved effo...https://math.codidact.com/posts/281009Why aren't $z_1=f(xy)$ and $z_2=f(x/y)$ functions of 2 variables?VLDRhttps://math.codidact.com/users/536962021-03-09T05:48:14Z2021-03-09T07:24:02Z<p><a href="https://math.stackexchange.com/a/738461/53259">Hagen von Eitzen answered</a> that $z_1, z_2$</p>
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<p>depend on only one variable - there's no comma between the parentheses...https://math.codidact.com/posts/281008If a repair on a vessel costs $2 and fully repaired vessel is worth $1, don't we need know worth of broken vessel to decide whether to repair?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:37:11Z2021-03-09T05:37:11Z<p>I divided all prices by 100,000 to simplify the question. The book beneath doesn't stipulate the current value of the vessel. But don't we need to know it, to calculate if it's worth repairing? ...https://math.codidact.com/posts/281007What kind of logic does “Logic in the Theory and Practice of Lawmaking” employ?Chgg Clouhttps://math.codidact.com/users/535642021-03-09T05:30:15Z2021-03-09T05:30:15Z<p>I stumbled on 2016 Springer book <a href="https://www.amazon.ca/Theory-Practice-Lawmaking-Michal-Araszkiewicz/dp/3319371460"><em>Logic in the Theory and Practice of Lawmaking</em></a>. I was cur...https://math.codidact.com/posts/280910Approximation of an elliptic integraltheabbiehttps://math.codidact.com/users/537872021-02-24T03:00:54Z2021-02-24T03:00:54Z<p>I used some nice guesswork to get this formula<br>
$$\int_0^{\frac{\pi}{2}}\sqrt{a^2\sin^2x+b^2\cos^2x}dx=\frac{ab\pi}{\left(a+b\right)\sin\left(\frac{a\pi}{a+b}\right)}$$<br>
The comparision ...https://math.codidact.com/posts/280851If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$?VLDRhttps://math.codidact.com/users/536962021-02-18T06:39:06Z2021-02-22T08:11:42Z<p>Did Stewart prove the titular result, also underlined in green in the image below, either by himself or as an exercise, in <em>Calculus Early Transcendentals</em> or in the normal version <em>Ca...https://math.codidact.com/posts/280842Does every divergence-free vector field arise as the curl of some vector field?Bahudari Ragamhttps://math.codidact.com/users/535162021-02-17T18:51:20Z2021-02-20T19:38:13Z<p>I was introduced to the concepts of gradient $\nabla f$, curl $\nabla \times F$ and divergence $\nabla \cdot F$ in an introductory course on calculus during my undergraduate studies. There I lea...https://math.codidact.com/posts/280848How can I visualize this Compound Interest Chart with indefinite integrals?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:36:11Z2021-02-18T20:09:22Z<p>How can I visualize <a href="https://betterexplained.com/articles/definitions-of-e-colorized/">the integrals below</a>? Can someone draw on the chart and point to where $\int 1 , dx = x$, $\int ...https://math.codidact.com/posts/280850How to visualize division as splitting Dividend into B equal “partial groups”, then rounding up A partial groups to get a full group?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:38:08Z2021-02-18T06:38:08Z<p>Because $10 \div \dfrac{4}{3}$ isn't an integer, <a href="https://old.reddit.com/r/learnmath/comments/ckv75r/why_multiplying_by_the_reciprocal_works_while/evurf0g/">I changed the numbers in thi...https://math.codidact.com/posts/280849How can I visualize dividing by a fraction as partial and full groups?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:37:44Z2021-02-18T06:37:44Z<p>I can't visualize all these "full groups" and "partial groups." Can someone please picture them? Please see my questions red in-line. Please don't use the identity that $\dfr...https://math.codidact.com/posts/280847How can I re-write or supplant “basis point” unambiguously?Chgg Clouhttps://math.codidact.com/users/535642021-02-18T06:29:34Z2021-02-18T06:31:16Z<p>For this question, please don't use or even allude to "basis point" in your rewording. <a href="https://www.investopedia.com/ask/answers/what-basis-point-bps/">Basis Point (BPS) Defini...https://math.codidact.com/posts/280704Why should a non-commutative operation even be called "multiplication"?theabbiehttps://math.codidact.com/users/537872021-02-08T10:18:35Z2021-02-14T19:56:50Z<p>As per my knowledge and what was taught in school,</p>
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<p>$a\times b$ is $a$ times $b$ or $b$ times $a$</p>
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<p>Obviously this is commutative as $a$ times $b$ and $...https://math.codidact.com/posts/280663How can I generalize a picture for the Mean Value Theorem to the Generalized MVT?VLDRhttps://math.codidact.com/users/536962021-02-04T02:22:15Z2021-02-11T22:16:44Z<p>How can I transmogrify this figure for the <a href="https://math.stackexchange.com/q/296176">Generalized MVT</a>? If $f$ and $g$ are continuous
on the closed interval $[a, b]$ and differentiabl...https://math.codidact.com/posts/280758In a SARIMA model, does the order in which I difference the time series matter?anoldmanintheseahttps://math.codidact.com/users/536472021-02-10T18:11:16Z2021-02-10T18:11:16Z<p>In a SARIMA model $\text{ARIMA}(p,d,q)\times(P,D,Q)_S$, we can write it as $$\Phi_P(B^S)\phi(B)(1-B^S)^D(1-B)^d x_t= \delta +\Theta_Q(B^S)\theta(B)w_t$$ where the capital letters are the paramet...https://math.codidact.com/posts/280741What story and one-digit Natural Numbers explain Bayes' Theorem chart most simply?DNBhttps://math.codidact.com/users/536282021-02-09T23:29:08Z2021-02-10T09:42:21Z<p>Some students have sniveled that most examples of Bayes' Theorem use non-integer numbers. I want to try a Bayes' Theorem chart that uses just single digit Natural Numbers $\le 9$. To complete th...https://math.codidact.com/posts/279026Classification of topological pointsceltschkhttps://math.codidact.com/users/80562020-11-08T20:51:50Z2021-02-02T22:52:43Z<p>I've recently dreamed up the following idea to classify topological points, and I'm wondering if this is or relates to a known concept. Here “topological point” refers to a point together with i...https://math.codidact.com/posts/280629Isn't it wrong to write that Indefinite Integral = Definite Integral with a variable in its Upper Limit?VLDRhttps://math.codidact.com/users/536962021-01-31T23:53:38Z2021-02-02T21:29:00Z<blockquote>
<p>$\int f(t) \; dt = \int_{t_0}^t f(s) \; ds \quad \text{ where $t_0$ is some convenient lower limit of integration.}$</p>
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<p>Isn't this wrong? Because LHS $\neq$ RH...https://math.codidact.com/posts/280572Does this generalization of path-connectedness also cover general connectedness?celtschkhttps://math.codidact.com/users/80562021-01-24T14:06:30Z2021-01-24T14:08:17Z<p>I've got the following idea to generalise path-connectedness:</p>
<p>Given a topological space $P$ and a subspace $S$, a space $X$ is $(P,S)$-connected if every continuous function $f:S\to X$ c...https://math.codidact.com/posts/280524Are there useful topologies on Cartesian products “in between” the product topology and the box topology?celtschkhttps://math.codidact.com/users/80562021-01-20T07:44:16Z2021-01-20T08:45:21Z<p>On the Cartesian product of topological spaces, there are two standard topologies: One is the product topology, the other is the box topology.</p>
<p>As is well known, the box topology is gener...https://math.codidact.com/posts/278332Cutting the squareGrovehttps://math.codidact.com/users/534522020-10-01T21:53:10Z2021-01-18T15:15:15Z<p>I don't know how simple/diffcult this is, but it has been on my mind for some time, and I haven't managed to do anything about it, so now I'll try here to see if anyone has any input.</p>
<p>Le...https://math.codidact.com/posts/280495Why does this definition of generalized forces work?JoshuaTShttps://math.codidact.com/users/532632021-01-16T00:57:56Z2021-01-16T03:53:49Z<p>I don't understand an equation in the discussion of Lagrangian Mechanics in the book <em>Mathematical Aspects of Classical and Celestial Mechanics</em> by Arnold <em>et al.</em> (section 1.2.1)....