https://math.codidact.com/categories/41/tags/3863.rssNew Posts Tagged 'probability' - MathematicsMathematics - Codidact2022-05-09T13:01:11Zhttps://math.codidact.com/posts/285432How to intuit p = Calvin's probability of winning each game independently = $1/2 \implies$ P(Calvin wins the match) = 1/2?DNBhttps://math.codidact.com/users/536282021-12-31T08:31:07Z2022-05-09T13:01:11Z<p>Please see the sentence beside my red line. The notion of a "sanity check" suggests that these resultant integers should be obvious, without calculation or contemplation. But why's it plain and ...https://math.codidact.com/posts/286292Why does the Dottie number $=\sqrt{1-\left(2\text I^{-1}_\frac12\left(\frac12,\frac32\right)-1\right)^2}$?Tyma Gaidashhttps://math.codidact.com/users/565582022-04-22T04:26:17Z2022-04-22T04:37:45Z<p><strong>Introduction:</strong></p>
<p>For some background information on the <a href="https://en.m.wikipedia.org/wiki/Dottie_number"><strong>Dottie Number D</strong></a>,</p>
<p>Cross posted f...https://math.codidact.com/posts/285670Why 1. multiply the number of independent options? 2. add the number of exclusive options?DNBhttps://math.codidact.com/users/536282022-01-24T03:31:04Z2022-02-27T05:05:31Z<p>Ironically, this textbook highlights understanding over memorization, but it doesn't expatiate the two WHY's in the question title!</p>
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<p>When faced with a series of independent ...https://math.codidact.com/posts/282608Out 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?DNBhttps://math.codidact.com/users/536282021-07-09T05:04:41Z2022-01-28T15:52:59Z<ol>
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<p>Please see the sentence alongside my red line below. Why does part (a) overcount part (b) by a factor of c?</p>
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<p>Scilicet, why aren't the answers to parts (a) and (b...https://math.codidact.com/posts/285443Why's Pr(the running total of a fair dice rolled repeatedly = n) = $\frac16 (p_{n-1} + p_{n-2} + p_{n-3} + p_{n-4} + p_{n-5} + p_{n-6})$?DNBhttps://math.codidact.com/users/536282022-01-01T05:01:33Z2022-01-17T09:18:39Z<p>Where did $\color{red}{p_n = \frac16 (p_{n-1} + p_{n-2} + p_{n-3} + p_{n-4} + p_{n-5} + p_{n-6})}$ spring from? The solution doesn't expatiate, and makes it appear out of the blue? Can you pleas...https://math.codidact.com/posts/285433Why's a 2-player game with "win by two" rule = Gambler's Ruin where each player starts with $2? Why N = 4?DNBhttps://math.codidact.com/users/536282021-12-31T08:41:57Z2022-01-17T08:54:19Z<ol>
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<p>Why can this problem "be thought of as a gambler's ruin where each player starts out
with $2"?</p>
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<p>Please see my red arrow. Why is this exponent 4? I quote op. cit....https://math.codidact.com/posts/285431If C = Calvin wins the match, and $X \thicksim Bin(2, p) =$ how many of the first 2 games he wins — then why P(C|X = 1) = P(C)?DNBhttps://math.codidact.com/users/536282021-12-31T08:24:45Z2022-01-05T11:14:44Z<p>The author's solution doesn't expatiate why $\color{red}{P(C|X = 1) = P(C)}$? <a href="https://math.stackexchange.com/q/4288685">This similar question on Math Stack Exchange</a> has 0 answers, a...https://math.codidact.com/posts/285430Intuitively, why does $p$ vary inversely with $P(C_3 \mid D_2)$? But directly with $P(C_2 \mid D_3)$?DNBhttps://math.codidact.com/users/536282021-12-31T07:29:04Z2021-12-31T15:05:54Z<p><strong>I'm seeking merely intuition here</strong> — NOT about the algebra that I know how to execute. Please see the fractions colored in red and orange at the bottom.</p>
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<li>$P(C_3 \...https://math.codidact.com/posts/285434"the people who move have a percentage of Democrats which is between these two values" — How does this furnish intuition for Simpson's Paradox?DNBhttps://math.codidact.com/users/536282021-12-31T08:53:03Z2021-12-31T08:55:07Z<p>Why does the phrase colored in red below ($\color{red}{\text{the people who move have a percentage of Democrats which is between these two values}}$) matter? How does it assist with intuiting t...https://math.codidact.com/posts/285429For (a variation) of the Monty Hall problem, what permits you to repaint doors 2 and 3? What permits you to replace p with $1 - p$?DNBhttps://math.codidact.com/users/536282021-12-31T07:07:53Z2021-12-31T08:34:35Z<p>Please see the text colored in red at the bottom.</p>
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<p>Why can you haughtily just — and what legitimizes you to — "[i]magine repainting doors 2 and 3, reversing which is called wh...https://math.codidact.com/posts/282603Without calculations, how can you visualize "that half the squares are white and half are black"?DNBhttps://math.codidact.com/users/536282021-07-09T03:50:39Z2021-12-30T23:34:58Z<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/285364In general, does $\color{forestgreen}{P(A|M)} + \color{red}{P(B|M)} = 1$?DNBhttps://math.codidact.com/users/536282021-12-23T09:08:25Z2021-12-30T22:06:30Z<p>In this question, $\color{forestgreen}{P(A|M)} + \color{red}{P(B|M)} = 1$. But the author's solution didn't annunciate this, and doesn't unfurl how to compute $\color{red}{P(B|M)}$. Can I simply...https://math.codidact.com/posts/285365Acceptable, usual to write $\ge 2$ pipes simultaneously?DNBhttps://math.codidact.com/users/536282021-12-23T09:24:56Z2021-12-26T09:47:28Z<p><em><strong>I'm NOT asking for the solution to this exercise that's publicly accessible.</strong></em> Rather, pls see the green and red underlines. If I apply the author's green definition to t...https://math.codidact.com/posts/285371How does $P(C > D \mid C = 2) \neq P(C > D \mid C \neq 2)$ prove that B > C depends on C > D?DNBhttps://math.codidact.com/users/536282021-12-24T07:00:13Z2021-12-24T07:00:33Z<p>I grok that $\color{limegreen}{P(C > D \mid C = 2) = P(D = 1 \mid C = 6) = 1/2}$, and $\color{red}{P(C > D \mid C \neq 2) = P(C > D \mid C = 6) = 1}$. But I don't grok the last sentenc...https://math.codidact.com/posts/282737Probability of students given there are groupsPranav_M99https://math.codidact.com/users/541902021-07-21T16:12:05Z2021-11-02T14:53:19Z<p>A group of 60 second graders is to be randomly assigned to two classes of 30 each.
(The random assignment is ordered by the school district to ensure against any
bias.) Five of the second grad...https://math.codidact.com/posts/282643If A & B are joint, can Arby recoup some of his loss only when $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$?DNBhttps://math.codidact.com/users/536282021-07-11T06:07:51Z2021-09-18T08:18:31Z<ol>
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<p>Please see the sentence alongside the red line below, but the authors didn't write this sentence for the first case ( $P_{Arby}(A \cup B) < P_{Arby}(A) + P_{Arby}(B)$). Thus if A ...https://math.codidact.com/posts/282645If Alice must've have classes on at least 2 days, why do you need the intersection of 3 $A_i^C$'s?DNBhttps://math.codidact.com/users/536282021-07-11T06:42:50Z2021-09-18T06:01:51Z<p>Can someone please rectify my MathJax? Please see the red phrase below.</p>
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<p>The question itself never touts or postulates outright that Alice "must have classes on at least 2 da...https://math.codidact.com/posts/283260How would you vaticinate to $-w_k$ from both sides of $w_{k + 1} = \dfrac{w_k - (1 - p)w_{k - 1}}{p}$?DNBhttps://math.codidact.com/users/536282021-08-10T06:25:17Z2021-09-16T16:22:40Z<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/283252Which vertical line signifies "putting the cutoff for a positive result at a very low level"?DNBhttps://math.codidact.com/users/536282021-08-09T23:44:39Z2021-09-12T05:14:47Z<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/283111Why's the true positive rate termed Sensitivity and true negative rate Specificity, not vice versa? DNBhttps://math.codidact.com/users/536282021-08-05T02:11:34Z2021-09-02T22:39:51Z<ol>
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<p>To wit, what's "Sensitive" about True Positive Rates, and "Specific" about True Negative Rates?</p>
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<p>Why weren't these <a href="https://en.wikipedia.org/wiki/Semantic...https://math.codidact.com/posts/282604Why aren't the "21 possibilities here" NOT equally likely? DNBhttps://math.codidact.com/users/536282021-07-09T03:59:27Z2021-09-02T14:15:09Z<p>Please see the last sentence below, that I highlighted in red.</p>
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<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/282888How can I visualize the Law of Total Probability with extra conditioning?DNBhttps://math.codidact.com/users/536282021-07-28T00:06:49Z2021-09-02T14:14:13Z<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>
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<p><strong>Theorem 2....https://math.codidact.com/posts/282889How do you calculate $P(X = n|G), P(X = n|G^C)$ by the Law of Total Probability, with extra conditioning? DNBhttps://math.codidact.com/users/536282021-07-28T02:15:02Z2021-09-02T14:12:22Z<ol>
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<p>Please see $P(U|G)$ and $P(U|G^C)$ below, beside my red line. Can you please expound these calculations?</p>
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<p>How was $s$ computed in both equations?</p>
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</o...https://math.codidact.com/posts/282614Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"?DNBhttps://math.codidact.com/users/536282021-07-09T06:38:52Z2021-09-02T14:11:53Z<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/283887Does "both girls, ≥ 1 winter girl" = "both girls, ≥ 1 winter child" let you generalize the problem statement that postulated both children as girls? DNBhttps://math.codidact.com/users/536282021-09-01T08:40:20Z2021-09-01T08:40:58Z<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/283886How would you intuit, and soothsay to rewrite, "both girls, ≥ 1 winter girl" as "both girls, ≥ 1 winter child"?DNBhttps://math.codidact.com/users/536282021-09-01T08:35:52Z2021-09-01T08:37:39Z<ol>
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<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>
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<...https://math.codidact.com/posts/283388How do I prove Simpson's Paradox, scilicet $P(A|B) > P(A|B^C)$?DNBhttps://math.codidact.com/users/536282021-08-14T08:22:59Z2021-08-27T09:10:11Z<p>$\forall a,b,c,d > 0, a<b, c<d \implies$ <a href="https://math.stackexchange.com/q/3632752">$0 \le ab < cd$</a>.</p>
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<p>The blue, orange and purple inequalities below ar...https://math.codidact.com/posts/283255After $n - 2$ unchosen doors are opened, how does the probability of the $n - 2$ unchosen doors "shift" or "transfer" to the lone unchosen door?DNBhttps://math.codidact.com/users/536282021-08-10T00:43:39Z2021-08-24T06:08:04Z<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/283118Why would skyrocketing the numbers of doors help laypeople intuit the Monty Hall Problem? DNBhttps://math.codidact.com/users/536282021-08-05T04:27:25Z2021-08-20T23:17:49Z<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/282942Besides guaranteeing 0 ≤ P(A ∩ B) ≤ 1, why can't Independence be defined as P(A ∩ B) = P(A) +,-, or ÷ P(B)? DNBhttps://math.codidact.com/users/536282021-07-29T08:02:00Z2021-08-20T23:10:11Z<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/283295In $w_{k + 1} - w_k = (\frac{1 - p}{p})^{exponent}(w_1 - w_0)$, why isn't exponent $k + 1$?DNBhttps://math.codidact.com/users/536282021-08-11T08:23:25Z2021-08-14T13:43:51Z<p>Please see the $r^k$ underlined in red, which is $(\frac{1 - p}{p})^k$ as defined by the green underlines.</p>
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<p>How do you deduce that the exponent must be $k$? Why isn't the expo...https://math.codidact.com/posts/283387Why "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)?DNBhttps://math.codidact.com/users/536282021-08-14T07:33:58Z2021-08-14T07:33:58Z<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/283371How can I exploit symmetry to intuit P(A wins) + P(B wins) = 1, without performing algebra? DNBhttps://math.codidact.com/users/536282021-08-13T06:55:38Z2021-08-13T06:55:38Z<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/283297How 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 $p \neq 1 - p$?DNBhttps://math.codidact.com/users/536282021-08-11T08:35:07Z2021-08-11T08:36:43Z<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/283115How does P(Monty opens door 2) = P(Monty opens door 3), and $P(\text{get car}|M_2)P(M_2) = P(\text{get car}|M_3)P(M_3)$?DNBhttps://math.codidact.com/users/536282021-08-05T04:07:10Z2021-08-10T07:20:03Z<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/283254How to visualize multiplication in the Odds form of Bayes's Theorem?DNBhttps://math.codidact.com/users/536282021-08-10T00:27:13Z2021-08-10T00:28:38Z<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/283253How to visualize division in the Odds form of Bayes's Theorem? DNBhttps://math.codidact.com/users/536282021-08-10T00:04:16Z2021-08-10T00:28:21Z<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/282771How can "information about the birth season" bring "at least one is a girl" closer to "a specific one is a girl"? DNBhttps://math.codidact.com/users/536282021-07-24T07:29:34Z2021-08-06T18:20:43Z<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/283116Intuitively, why would organisms — that after one minute, will either die, split into two, or stay the same, with equal probability — all die ultimately? DNBhttps://math.codidact.com/users/536282021-08-05T04:17:35Z2021-08-05T07:50:35Z<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/283113In 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?DNBhttps://math.codidact.com/users/536282021-08-05T03:53:12Z2021-08-05T04:05:52Z<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>
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<li>Correct me if I'm wrong, but the game show didn't ...https://math.codidact.com/posts/283112If a 2nd test's independent from the 1st test, then why does $\frac{0.95}{0.05}$ figure twice in $\frac{P(D|T_1)}{P(D^C|T_1)}\frac{P(T_2|D,T_1)}{P(T_2|D^C,T_1)}$? DNBhttps://math.codidact.com/users/536282021-08-05T03:39:12Z2021-08-05T03:41:26Z<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/283110How do these 3 bell curves of Likelihood, Posterior, Prior pictorialize the Odds form of Bayes' rule? DNBhttps://math.codidact.com/users/536282021-08-05T01:45:09Z2021-08-05T01:46:23Z<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/282943Intuitively, why does A, B independent $\iff$ A, $B^C$ independent $\iff A^C, B^C$ independent?DNBhttps://math.codidact.com/users/536282021-07-29T08:10:47Z2021-07-29T11:03:19Z<blockquote>
<h3>Proposition 2.5.3.</h3>
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<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/282605Intuitively, if you pick k out of n objects singly without replacement, why's the number of possible outcomes NOT $n(n-1) \dots [(n-(k - 1)]\color{red}{(n - k)}$?DNBhttps://math.codidact.com/users/536282021-07-09T04:08:51Z2021-07-28T04:05:35Z<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/282873In the Lost Boarding Pass Probability Problem, why couldn't Passengers 2-99 sit in Seats 1 or 100, before Passenger 100 boards? DNBhttps://math.codidact.com/users/536282021-07-27T05:41:40Z2021-07-27T23:56:41Z<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/282666Why isn't the probability of being void in 3 specifi�c suits $\frac{1/13}{\dbinom{52}{13}}$?DNBhttps://math.codidact.com/users/536282021-07-13T06:55:04Z2021-07-13T06:55:04Z<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/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/535642021-06-05T23:38:20Z2021-07-11T15:05:29Z<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/282606If k = 1, why $n(n-1) \dots \color{red}{(n-k+1)} = n$?DNBhttps://math.codidact.com/users/536282021-07-09T04:21:14Z2021-07-09T07:47:58Z<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>
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...https://math.codidact.com/posts/282613Why shouldn't the Bose-Einstein value be used to calculate birthday probabilities?DNBhttps://math.codidact.com/users/536282021-07-09T06:29:58Z2021-07-09T06:29:58Z<p>Can you please expound and simplify the embolden phrase below?</p>
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<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/282612You're sampling k people from a population of size n one at a time, with replacement and with equal probabilities. Order or not? DNBhttps://math.codidact.com/users/536282021-07-09T05:59:52Z2021-07-09T05:59:52Z<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...