https://math.codidact.com/categories/41/tags/3863.rss New Posts Tagged 'probability' - Mathematics Mathematics - Codidact 2021-09-18T08:18:31Z https://math.codidact.com/posts/282643 If A & B are joint, can Arby recoup some of his loss only when $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$? DNB‭ https://math.codidact.com/users/53628 2021-07-11T06:07:51Z 2021-09-18T08:18:31Z <ol> <li> <p>Please see the sentence alongside the red line below, but the authors didn't write this sentence for the first case ( $P_{Arby}(A \cup B) &lt; P_{Arby}(A) + P_{Arby}(B)$). Thus if A ... https://math.codidact.com/posts/282645 If Alice must've have classes on at least 2 days, why do you need the intersection of 3 $A_i^C$'s? DNB‭ https://math.codidact.com/users/53628 2021-07-11T06:42:50Z 2021-09-18T06:01:51Z <p>Can someone please rectify my MathJax? Please see the red phrase below.</p> <ol> <li> <p>The question itself never touts or postulates outright that Alice "must have classes on at least 2 da... https://math.codidact.com/posts/283260 How would you vaticinate to $-w_k$ from both sides of $w_{k + 1} = \dfrac{w_k - (1 - p)w_{k - 1}}{p}$? DNB‭ https://math.codidact.com/users/53628 2021-08-10T06:25:17Z 2021-09-16T16:22:40Z <p>This question appeared on my pop quiz last week. I got 0%. I achieved everything until the green equation, then I didn't know how to proceed. After reading this solution, I see that you must iso... https://math.codidact.com/posts/283252 Which vertical line signifies "putting the cutoff for a positive result at a very low level"? DNB‭ https://math.codidact.com/users/53628 2021-08-09T23:44:39Z 2021-09-12T05:14:47Z <p>The author, Karen Stewart <a href="https://www.linkedin.com/in/karen-steward-89036a24/">MA Natural Sciences (Univ. of Cambridge) PhD Veterinary Microbiology (Cambridge)</a>, refers to "a purple ... https://math.codidact.com/posts/283111 Why's the true positive rate termed Sensitivity and true negative rate Specificity, not vice versa? DNB‭ https://math.codidact.com/users/53628 2021-08-05T02:11:34Z 2021-09-02T22:39:51Z <ol> <li> <p>To wit, what's "Sensitive" about True Positive Rates, and "Specific" about True Negative Rates?</p> </li> <li> <p>Why weren't these <a href="https://en.wikipedia.org/wiki/Semantic... https://math.codidact.com/posts/282604 Why aren't the "21 possibilities here" NOT equally likely? DNB‭ https://math.codidact.com/users/53628 2021-07-09T03:59:27Z 2021-09-02T14:15:09Z <p>Please see the last sentence below, that I highlighted in red.</p> <hr> <p>Example $1.4 .5$ (Ice cream cones). Suppose you are buying an ice cream cone. You can choose whether to have a cake c... https://math.codidact.com/posts/282603 Without calculations, how can you visualize "that half the squares are white and half are black"? DNB‭ https://math.codidact.com/users/53628 2021-07-09T03:50:39Z 2021-09-02T14:14:35Z <p>Please see the 2nd para. below alongside my red highlighted words. I can't "[i]magine rotating the chessboard 90 degrees clockwise." I can't visualize how "all the positions that had a white squ... https://math.codidact.com/posts/282888 How can I visualize the Law of Total Probability with extra conditioning? DNB‭ https://math.codidact.com/users/53628 2021-07-28T00:06:49Z 2021-09-02T14:14:13Z <p>How can I pictorialized this Theorem 2.4.3? As you can see below, I edited a picture by drawing E inside B.. Is my edit correct? Can my edit be improved?</p> <blockquote> <p><strong>Theorem 2.... https://math.codidact.com/posts/282889 How do you calculate $P(X = n|G), P(X = n|G^C)$ by the Law of Total Probability, with extra conditioning? DNB‭ https://math.codidact.com/users/53628 2021-07-28T02:15:02Z 2021-09-02T14:12:22Z <ol> <li> <p>Please see $P(U|G)$ and $P(U|G^C)$ below, beside my red line. Can you please expound these calculations?</p> </li> <li> <p>How was $s$ computed in both equations?</p> </li> </o... https://math.codidact.com/posts/282614 Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"? DNB‭ https://math.codidact.com/users/53628 2021-07-09T06:38:52Z 2021-09-02T14:11:53Z <p>Because you "first choose the k team members and then choose one of time to be captain", shouldn’t the RHS be $\color{red}{k\dbinom{k}{1}}$? The captain is chosen from the $k$ team members alre... https://math.codidact.com/posts/283887 Does "both girls, ≥ 1 winter girl" = "both girls, ≥ 1 winter child" let you generalize the problem statement that postulated both children as girls? DNB‭ https://math.codidact.com/users/53628 2021-09-01T08:40:20Z 2021-09-01T08:40:58Z <p>What does "the fact that "both girls, at least one winter girl" is the same event as both girls, at least one winter child"" imply about the problem statement below that posited merely girls? I... https://math.codidact.com/posts/283886 How would you intuit, and soothsay to rewrite, "both girls, ≥ 1 winter girl" as "both girls, ≥ 1 winter child"? DNB‭ https://math.codidact.com/users/53628 2021-09-01T08:35:52Z 2021-09-01T08:37:39Z <ol> <li> <p>Please see the red underline. How can you intuit that "both girls, at least one winter girl" as "both girls, at least one winter child"? I ask this for my 15 y.o.</p> </li> <li> <... https://math.codidact.com/posts/283388 How do I prove Simpson's Paradox, scilicet $P(A|B) > P(A|B^C)$? DNB‭ https://math.codidact.com/users/53628 2021-08-14T08:22:59Z 2021-08-27T09:10:11Z <p>$\forall a,b,c,d &gt; 0, a&lt;b, c&lt;d \implies$ <a href="https://math.stackexchange.com/q/3632752">$0 \le ab &lt; cd$</a>.</p> <ol> <li> <p>The blue, orange and purple inequalities below ar... https://math.codidact.com/posts/283255 After $n - 2$ unchosen doors are opened, how does the probability of the $n - 2$ unchosen doors "shift" or "transfer" to the lone unchosen door? DNB‭ https://math.codidact.com/users/53628 2021-08-10T00:43:39Z 2021-08-24T06:08:04Z <p>The second quotation below uses the verb "shift" to describe how Monty Hall's opening the 98 unchosen doors (revealing a goat each) ""shifts" [boldening mine] that 99/100 chance to door #100"? T... https://math.codidact.com/posts/283118 Why would skyrocketing the numbers of doors help laypeople intuit the Monty Hall Problem? DNB‭ https://math.codidact.com/users/53628 2021-08-05T04:27:25Z 2021-08-20T23:17:49Z <p>Alas, it isn't clear to me that <strong>it becomes clear that the probabilities are not 50-50 for the two unopened doors</strong>. Had I never seen this exercise or problem, even if there were ... https://math.codidact.com/posts/282942 Besides guaranteeing 0 ≤ P(A ∩ B) ≤ 1, why can't Independence be defined as P(A ∩ B) = P(A) +,-, or ÷ P(B)? DNB‭ https://math.codidact.com/users/53628 2021-07-29T08:02:00Z 2021-08-20T23:10:11Z <p>I know that $0 \le P(A \cap B) \le 1$ will be violated if $P(A \cap B) = P(A) +,-,\text{or} ÷ P (B)$. I'm not asking about or challenging this reason.</p> <p>But what are the other reasons agai... https://math.codidact.com/posts/283295 In $w_{k + 1} - w_k = (\frac{1 - p}{p})^{exponent}(w_1 - w_0)$, why isn't exponent $k + 1$? DNB‭ https://math.codidact.com/users/53628 2021-08-11T08:23:25Z 2021-08-14T13:43:51Z <p>Please see the $r^k$ underlined in red, which is $(\frac{1 - p}{p})^k$ as defined by the green underlines.</p> <ol> <li> <p>How do you deduce that the exponent must be $k$? Why isn't the expo... https://math.codidact.com/posts/283387 Why "only 1/10,000 men with wives they abuse subsequently murder them" ≠ P(A|G,M) & "50% of husbands who murder their wives abused them” ≠ P(G|A)? DNB‭ https://math.codidact.com/users/53628 2021-08-14T07:33:58Z 2021-08-14T07:33:58Z <p>All emboldings are mine. See my red side line — the solution identifies <strong>"only 1 in 10,000 men with wives they abuse subsequently murder their wives"</strong> (in the problem statement) ... https://math.codidact.com/posts/283371 How can I exploit symmetry to intuit P(A wins) + P(B wins) = 1, without performing algebra? DNB‭ https://math.codidact.com/users/53628 2021-08-13T06:55:38Z 2021-08-13T06:55:38Z <p>I'm not asking about algebra that I can execute. How can I intuit P(A wins) + P(B wins) = 1 most quickly, without algebra? Any 15 year old can calculate that for the</p> <p>$p = 1/2$ case, $\df... https://math.codidact.com/posts/283297 How to vaticinate calculating$w_{k + 1} = w_1(1 + \dfrac{1- p}{p} + \dots + [\dfrac{1- p}{p}]^k)$separately for$p = 1 - p$and$p \neq 1 - p$? DNB‭ https://math.codidact.com/users/53628 2021-08-11T08:35:07Z 2021-08-11T08:36:43Z <p>Please see the orange, green, red underlines. 1. Why does$\color{limegreen}{w_{k + 1} = w_1(1 + \dfrac{1- p}{p} + \dots + p\dfrac{1- p}{p}]^k)}$beget the two cases of$\color{red}p = 1 - p$a... https://math.codidact.com/posts/283115 How does P(Monty opens door 2) = P(Monty opens door 3), and$P(\text{get car}|M_2)P(M_2) = P(\text{get car}|M_3)P(M_3)$? DNB‭ https://math.codidact.com/users/53628 2021-08-05T04:07:10Z 2021-08-10T07:20:03Z <p><a href="https://math.codidact.com/posts/283113">"Monty, who knows where the car is, then opens one of the two remaining doors. The door he opens always has a goat behind it (he never reveals th... https://math.codidact.com/posts/283254 How to visualize multiplication in the Odds form of Bayes's Theorem? DNB‭ https://math.codidact.com/users/53628 2021-08-10T00:27:13Z 2021-08-10T00:28:38Z <p>Here I'm asking solely about the circle pictograms. Please eschew numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT asking about how to multipl... https://math.codidact.com/posts/283253 How to visualize division in the Odds form of Bayes's Theorem? DNB‭ https://math.codidact.com/users/53628 2021-08-10T00:04:16Z 2021-08-10T00:28:21Z <p>Here I'm asking solely about the circle pictograms. Please eschew referring to, or using, numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT ask... https://math.codidact.com/posts/282771 How can "information about the birth season" bring "at least one is a girl" closer to "a specific one is a girl"? DNB‭ https://math.codidact.com/users/53628 2021-07-24T07:29:34Z 2021-08-06T18:20:43Z <p>Please see the sentences beside my red highlighted words. I don't understand how "Conditioning on more and more specific information brings the probability closer and closer to$1/2$"?</p> <hr... https://math.codidact.com/posts/283116 Intuitively, why would organisms — that after one minute, will either die, split into two, or stay the same, with equal probability — all die ultimately? DNB‭ https://math.codidact.com/users/53628 2021-08-05T04:17:35Z 2021-08-05T07:50:35Z <p>I have no questions on the solution or the algebra, but even after re-reading the solution, I still can't fathom or intuit why$P(D) = 1$from the problem statement. Even now, I couldn't have di... https://math.codidact.com/posts/283113 In the Monty Hall problem, why can you just assume the contestant picked door 1? Why are you entitled to relabel the doors, or rewrite this solution with the door numbers permuted? DNB‭ https://math.codidact.com/users/53628 2021-08-05T03:53:12Z 2021-08-05T04:05:52Z <p>My bafflement ought be obvious. 1. A contestant could've picked doors 2, 3. So you can't just assume he picked door 1.</p> <ol start="2"> <li>Correct me if I'm wrong, but the game show didn't ... https://math.codidact.com/posts/283112 If a 2nd test's independent from the 1st test, then why does$\frac{0.95}{0.05}$figure twice in$\frac{P(D|T_1)}{P(D^C|T_1)}\frac{P(T_2|D,T_1)}{P(T_2|D^C,T_1)}$? DNB‭ https://math.codidact.com/users/53628 2021-08-05T03:39:12Z 2021-08-05T03:41:26Z <p>The problem statement postulates that <strong>"The new test is independent of the original test (given his disease status)"</strong>. So where did the two$\frac{0.95}{ 0.05}$, that I underlin... https://math.codidact.com/posts/283110 How do these 3 bell curves of Likelihood, Posterior, Prior pictorialize the Odds form of Bayes' rule? DNB‭ https://math.codidact.com/users/53628 2021-08-05T01:45:09Z 2021-08-05T01:46:23Z <p>I learn best visually, and I found these <a href="https://www.thebottomline.org.uk/blog/ebm/bayesian-statistics/">graph</a>. 1. Does it furnish intuition on Theorem 2.3.5 below?</p> <ol start="... https://math.codidact.com/posts/282943 Intuitively, why does A, B independent$\iff$A,$B^C$independent$\iff A^C, B^C$independent? DNB‭ https://math.codidact.com/users/53628 2021-07-29T08:10:47Z 2021-07-29T11:03:19Z <blockquote> <h3>Proposition 2.5.3.</h3> </blockquote> <blockquote> <p>If A and B are independent, <a href="https://math.stackexchange.com/a/2922539">then A and$B^C$are independent,$A^C$an... https://math.codidact.com/posts/282605 Intuitively, if you pick k out of n objects singly without replacement, why's the number of possible outcomes NOT$n(n-1) \dots [(n-(k - 1)]\color{red}{(n - k)}$? DNB‭ https://math.codidact.com/users/53628 2021-07-09T04:08:51Z 2021-07-28T04:05:35Z <p>I know that$\color{limegreen}{(n-k+1)} \equiv (n - (k - 1))$. But whenever I contemplate choosing k from n objects singly without replacement, I keep muffing the number of possible outcomes as ... https://math.codidact.com/posts/282873 In the Lost Boarding Pass Probability Problem, why couldn't Passengers 2-99 sit in Seats 1 or 100, before Passenger 100 boards? DNB‭ https://math.codidact.com/users/53628 2021-07-27T05:41:40Z 2021-07-27T23:56:41Z <p>Although <a href="https://www.linkedin.com/in/tanae-rao-05a650198/?originalSubdomain=au">Tanae Rao</a> was just a high school graduate when he wrote <a href="https://towardsdatascience.com/the-l... https://math.codidact.com/posts/282666 Why isn't the probability of being void in 3 specifi�c suits$\frac{1/13}{\dbinom{52}{13}}$? DNB‭ https://math.codidact.com/users/53628 2021-07-13T06:55:04Z 2021-07-13T06:55:04Z <p>Kindly see the sentence UNDER the red line below. Why isn't the probability being void in 3 specific suits$\frac{1/13}{\dbinom{52}{13}}$? As "the probability of being void in 3 specific suits" ... https://math.codidact.com/posts/282046 Why 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 Clou‭ https://math.codidact.com/users/53564 2021-06-05T23:38:20Z 2021-07-11T15:05:29Z <p>Please see the embolden sentence below. Assume that I'm risk adverse and &quot;prefer the known chance over the unknown&quot;. Why's it irrational for me to choose A?</p> <blockquote> <p> ... https://math.codidact.com/posts/282606 If k = 1, why$n(n-1) \dots \color{red}{(n-k+1)} = n$? DNB‭ https://math.codidact.com/users/53628 2021-07-09T04:21:14Z 2021-07-09T07:47:58Z <p>Please see the boldened sentence below. I write out the LHS$= n(n-1) \dots (n-[k-3])(n-[k-2])\color{red}{(n-[k-1])}$. Then$LHS|_ {k = 1} = n(n-1) \dots (n+2)(n+1) \neq n$.</p> <blockquote> ... https://math.codidact.com/posts/282613 Why shouldn't the Bose-Einstein value be used to calculate birthday probabilities? DNB‭ https://math.codidact.com/users/53628 2021-07-09T06:29:58Z 2021-07-09T06:29:58Z <p>Can you please expound and simplify the embolden phrase below?</p> <blockquote> <p>As another example, with n = 365 days in a year and k people, how many possible unordered birthday lists ar... https://math.codidact.com/posts/282612 You're sampling k people from a population of size n one at a time, with replacement and with equal probabilities. Order or not? DNB‭ https://math.codidact.com/users/53628 2021-07-09T05:59:52Z 2021-07-09T05:59:52Z <p>If you're sampling k people from a population of size n one at a time, with replacement and with equal probabilities, then why does it matter whether your samples are ordered? The quotation belo... https://math.codidact.com/posts/282602 "A occurred" vs. "something must happen" DNB‭ https://math.codidact.com/users/53628 2021-07-09T03:44:45Z 2021-07-09T05:40:00Z <ol> <li> <p>Why doesn't "Something must happen" mean$s_{actual} \in A$?</p> </li> <li> <p>Scilicet, doesn't "A occurs" mean the same thing as "something must happen"? Something must happen... https://math.codidact.com/posts/282608 Out of 4 people, why does ways to choose a 2-person committee overcount by 2 the ways to divide the 4 into 2 teams of 2? DNB‭ https://math.codidact.com/users/53628 2021-07-09T05:04:41Z 2021-07-09T05:04:41Z <ol> <li> <p>Please see the sentence alongside my red line below. Why does part (a) overcount part (b) by a factor of c?</p> </li> <li> <p>Scilicet, why aren't the answers to parts (a) and (b... https://math.codidact.com/posts/282607 Explain to a 9 year old — To count each possibility c times, why divide by c? Why not subtract by c? DNB‭ https://math.codidact.com/users/53628 2021-07-09T04:33:28Z 2021-07-09T04:33:28Z <p>Please see the embolded phrase below. How can you explain to a 9 year old why you</p> <ol> <li> <p>must divide by$c$?</p> </li> <li> <p>can't subtract by$c$?</p> </li> </ol> <blockquo... https://math.codidact.com/posts/280168 How can I deduce which operation removes redundacies? DNB‭ https://math.codidact.com/users/53628 2020-12-30T03:11:56Z 2021-07-09T04:33:04Z <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> <ol> <li>How can I deduce which operation ought ... https://math.codidact.com/posts/281987 What story and two-digit Natural Numbers best fit Bayes' Theorem chart? DNB‭ https://math.codidact.com/users/53628 2021-06-02T07:36:40Z 2021-07-06T03:09:40Z <p>To complete the table below most comfortably for teenagers,</p> <ol> <li> <p>what are the simplest stories?</p> </li> <li> <p>what natural numbers$\le 99$contrast the base rate fallacy t... https://math.codidact.com/posts/282061 How 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 Clou‭ https://math.codidact.com/users/53564 2021-06-06T23:45:21Z 2021-06-10T23:13:26Z <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)$&gt; 1/5... https://math.codidact.com/posts/282055 How's the outcome of airplane crashes almost always certain, but the probability is not? Chgg Clou‭ https://math.codidact.com/users/53564 2021-06-06T08:26:15Z 2021-06-07T07:49:58Z <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> <ol> <li>Isn't the outcome of... https://math.codidact.com/posts/282060 Why isn't $\Pr(diseased|+ test) = \dfrac{\text{number of true positives}}{\text{number of false positives}}$? Chgg Clou‭ https://math.codidact.com/users/53564 2021-06-06T23:40:32Z 2021-06-06T23:41:45Z <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/282049 Why paradoxical to accept \$3, and avoid betting on Dushanbe's temperature for a prize of \$10? Chgg Clou‭ https://math.codidact.com/users/53564 2021-06-06T06:58:34Z 2021-06-06T23:07:58Z <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/282056 What does "individuals discount the future in a constant manner" mean? Chgg Clou‭ https://math.codidact.com/users/53564 2021-06-06T08:30:00Z 2021-06-06T08:30:00Z <ol> <li> <p>Is the embold phrase referring to Present Value?</p> </li> <li> <p>But how does Present Value relate to the example below with the &quot;online bookstore&quot;?</p> </li> </ol> ... https://math.codidact.com/posts/282052 If 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 Clou‭ https://math.codidact.com/users/53564 2021-06-06T07:08:55Z 2021-06-06T07:08:55Z <p>The emboldened sentences feel contradictory. On one hand, &quot;they are equal reductions in the probability of death&quot;. On the other, &quot;money is less valuable in the two-bullet case sin... https://math.codidact.com/posts/282051 Why 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 Clou‭ https://math.codidact.com/users/53564 2021-06-06T07:03:16Z 2021-06-06T07:04:52Z <p>Can you please expound the embold phrase below? Can this be calculated?</p> <blockquote> <p>      It is notoriously difficult, however, to assess what one’s prior assessment of the risk would ... https://math.codidact.com/posts/282014 How 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 Clou‭ https://math.codidact.com/users/53564 2021-06-04T04:43:32Z 2021-06-05T14:49:27Z <p>When I first saw inequalities (1) and (2) below, I quantified them as:</p> <p>$\color{red}{\text{1.1. married people &lt; all smokers}}$</p> <p>\$\color{red}{\text{2.1. and married smokers &lt... https://math.codidact.com/posts/282020 What was Justice Scalia's mathematical mistake in Penry v. Lynaugh (1989)? Chgg Clou‭ https://math.codidact.com/users/53564 2021-06-04T07:51:58Z 2021-06-05T02:05:16Z <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> <blockquote> <p>      But even grant...