https://math.codidact.com/categories/41/tags/3863.rss New Posts Tagged 'probability' - Mathematics Mathematics - Codidact 2022-05-09T13:01:11Z https://math.codidact.com/posts/285432 How to intuit p = Calvin's probability of winning each game independently = $1/2 \implies$ P(Calvin wins the match) = 1/2? DNB‭ https://math.codidact.com/users/53628 2021-12-31T08:31:07Z 2022-05-09T13:01:11Z <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/286292 Why does the Dottie number $=\sqrt{1-\left(2\text I^{-1}_\frac12\left(\frac12,\frac32\right)-1\right)^2}$? Tyma Gaidash‭ https://math.codidact.com/users/56558 2022-04-22T04:26:17Z 2022-04-22T04:37:45Z <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/285670 Why 1. multiply the number of independent options? 2. add the number of exclusive options? DNB‭ https://math.codidact.com/users/53628 2022-01-24T03:31:04Z 2022-02-27T05:05:31Z <p>Ironically, this textbook highlights understanding over memorization, but it doesn't expatiate the two WHY's in the question title!</p> <blockquote> <p>When faced with a series of independent ... 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 2022-01-28T15:52:59Z <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/285443 Why's Pr(the running total of a fair dice rolled repeatedly = n) = $\frac16 (p_{n-1} + p_{n-2} + p_{n-3} + p_{n-4} + p_{n-5} + p_{n-6})$? DNB‭ https://math.codidact.com/users/53628 2022-01-01T05:01:33Z 2022-01-17T09:18:39Z <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/285433 Why's a 2-player game with "win by two" rule = Gambler's Ruin where each player starts with $2? Why N = 4? DNB‭ https://math.codidact.com/users/53628 2021-12-31T08:41:57Z 2022-01-17T08:54:19Z <ol> <li> <p>Why can this problem "be thought of as a gambler's ruin where each player starts out with$2"?</p> </li> <li> <p>Please see my red arrow. Why is this exponent 4? I quote op. cit.... https://math.codidact.com/posts/285431 If 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)? DNB‭ https://math.codidact.com/users/53628 2021-12-31T08:24:45Z 2022-01-05T11:14:44Z <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/285430 Intuitively, why does $p$ vary inversely with $P(C_3 \mid D_2)$? But directly with $P(C_2 \mid D_3)$? DNB‭ https://math.codidact.com/users/53628 2021-12-31T07:29:04Z 2021-12-31T15:05:54Z <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> <ol> <li>$P(C_3 \... https://math.codidact.com/posts/285434 "the people who move have a percentage of Democrats which is between these two values" — How does this furnish intuition for Simpson's Paradox? DNB‭ https://math.codidact.com/users/53628 2021-12-31T08:53:03Z 2021-12-31T08:55:07Z <p>Why does the phrase colored in red below ($\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/285429 For (a variation) of the Monty Hall problem, what permits you to repaint doors 2 and 3? What permits you to replace p with$1 - p$? DNB‭ https://math.codidact.com/users/53628 2021-12-31T07:07:53Z 2021-12-31T08:34:35Z <p>Please see the text colored in red at the bottom.</p> <ol> <li> <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/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 2021-12-30T23:34:58Z <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/285364 In general, does$\color{forestgreen}{P(A|M)} + \color{red}{P(B|M)} = 1$? DNB‭ https://math.codidact.com/users/53628 2021-12-23T09:08:25Z 2021-12-30T22:06:30Z <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/285365 Acceptable, usual to write$\ge 2$pipes simultaneously? DNB‭ https://math.codidact.com/users/53628 2021-12-23T09:24:56Z 2021-12-26T09:47:28Z <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/285371 How does$P(C > D \mid C = 2) \neq P(C > D \mid C \neq 2)$prove that B > C depends on C > D? DNB‭ https://math.codidact.com/users/53628 2021-12-24T07:00:13Z 2021-12-24T07:00:33Z <p>I grok that$\color{limegreen}{P(C &gt; D \mid C = 2) = P(D = 1 \mid C = 6) = 1/2}$, and$\color{red}{P(C &gt; D \mid C \neq 2) = P(C &gt; D \mid C = 6) = 1}$. But I don't grok the last sentenc... https://math.codidact.com/posts/282737 Probability of students given there are groups Pranav_M99‭ https://math.codidact.com/users/54190 2021-07-21T16:12:05Z 2021-11-02T14:53:19Z <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/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/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...