Activity for DNB
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #282642 |
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— | almost 3 years ago |
| Edit | Post #282642 |
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— | almost 3 years ago |
| Edit | Post #282642 | Initial revision | — | almost 3 years ago |
| Question | — |
$\sum_{k=0}^{n} \binom{n}{k}=2^{n} \overset{?}{\iff} \sum_{k=0}^{n} \binom{2n+1}{k}=2^{2n}$ Jack D'Aurizio narratively proved $\color{red}{\sum\limits{k=0}^{n} \binom{2n+1}{k}=2^{2n}}$. Is this red equation related, and can it be transmogrified, to $\color{limegreen}{\sum\limits{k=0}^{n} \binom{n}{k}=2^{n}}$? I started my attempt by substituting $n = m/2$, because the RHS of the green... (more) |
— | almost 3 years ago |
| Comment | Post #282615 |
You wrote "You started on the right track with $n(n - 1)\ldots(n - [k - 3])(n - [k - 2])(n - [k - 1])$". I just evaluated this expression at $k = 1$. Now do you understand "what the middle expression in that line is supposed to mean"? (more) |
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| Comment | Post #282616 |
The issue here appears to be that the English syntax differs from the order of the terms on the RHS? I misconstrued "we could first choose the k team members" as $k$, and "then choose one of them to be captain" as $\dbinom{n}{k}$. (more) |
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| Edit | Post #282605 |
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— | almost 3 years ago |
| Comment | Post #282614 |
I saved up these questions over a week. But I'll slow down as you ask. "I'm a bit uncertain about asking tons of questions from other sources" Huh? I see nothing with asking questions from renowned textbooks?
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| Edit | Post #282605 |
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— | almost 3 years ago |
| Comment | Post #282611 |
Absolutely not! Manners please? I couldn't copy and paste from the second website. (more) |
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| Edit | Post #282606 |
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| Edit | Post #282614 | Initial revision | — | almost 3 years ago |
| Question | — |
Why $\color{red}{k\dbinom{k}{1}} \neq$ "first choose the k team members and then choose one of time to be captain"? 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 already chosen. $\color{forestgreen}{k\dbinom{n}{k}}$ appears wrong to me, because this means that you're c... (more) |
— | almost 3 years ago |
| Edit | Post #282613 | Initial revision | — | almost 3 years ago |
| Question | — |
Why shouldn't the Bose-Einstein value be used to calculate birthday probabilities? Can you please expound and simplify the embolden phrase below? >As another example, with n = 365 days in a year and k people, how many possible unordered birthday lists are there? For example, for k = 3, we want to count lists like (May 1, March 31, April 11), where all permutations are consid... (more) |
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| Edit | Post #282612 | Initial revision | — | almost 3 years ago |
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You're sampling k people from a population of size n one at a time, with replacement and with equal probabilities. Order or not? 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 below doesn't expound the pros and cons of ordering your samples or not. >1.4.23. The Bose-Einstein result ... (more) |
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| Edit | Post #282611 | Initial revision | — | almost 3 years ago |
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What's the bijection between Stars and Bars and Integer Solutions to an Equality? The second quotation below keeps mentioning "bijection", but it never explicitly defines it. So what's the formula for that bijection? A story instead of stars and bars - Making Your Own Sense > On to the third problem. As I said earlier, many people teach students to reduce other problems to ... (more) |
— | almost 3 years ago |
| Comment | Post #282602 |
Thanks. I fixed my typo. (more) |
— | almost 3 years ago |
| Edit | Post #282602 |
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— | almost 3 years ago |
| Edit | Post #282609 | Initial revision | — | almost 3 years ago |
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Are Stars and Bars in Combinatorics related to the Fence Post Error? The bars in the lower picture look like fences. That's why Stars and Bars reminds me of Fence Post Error? >It is common to replace the balls with “stars”, and to call the separators “bars”, yielding the popular name of the technique. We have 5 stars, and 2 bars in our example: > |
— | almost 3 years ago |
| Edit | Post #282608 | Initial revision | — | almost 3 years ago |
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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? 1. Please see the sentence alongside my red line below. Why does part (a) overcount part (b) by a factor of c? 2. Scilicet, why aren't the answers to parts (a) and (b) the same? Whenever you choose a 2-person committee #1, the remaining unchosen 2 members automatically can form the 2-person comm... (more) |
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| Edit | Post #282607 | Initial revision | — | almost 3 years ago |
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Explain to a 9 year old — To count each possibility c times, why divide by c? Why not subtract by c? Please see the embolded phrase below. How can you explain to a 9 year old why you 1. must divide by $c$? 2. can't subtract by $c$? >### 1.4.2 Adjusting for overcounting >In many counting problems, it is not easy to directly count each possibility once and only once. If, however, we are... (more) |
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| Edit | Post #280168 |
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| Edit | Post #280168 |
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| Edit | Post #280168 |
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| Comment | Post #281319 |
Thanks. Does my edit [to my post] change your answer? (more) |
— | almost 3 years ago |
| Edit | Post #280168 |
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— | almost 3 years ago |
| Edit | Post #282606 | Initial revision | — | almost 3 years ago |
| Question | — |
If k = 1, why $n(n-1) \dots \color{red}{(n-k+1)} = n$? 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 $LHS| {k = 1} = n(n-1) \dots (n+2)(n+1) \neq n$. >### Theorem 1.4.8 (Sampling without replacement). >Consider n objects and making k choices from them, one at a time wi... (more) |
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| Edit | Post #282605 |
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| Edit | Post #282605 |
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| Edit | Post #282605 | Initial revision | — | almost 3 years ago |
| Question | — |
Intuitively, if you pick k out of n objects singly without replacement, why's the number of possible outcomes NOT $n(n-1) \dots [(n-(k - 1)]\color{red}{(n - k)}$? 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 $n(n-1) \dots \color{limegreen}(n-k+1)\color{red}{(n - k)}$. I bungled by adding the unnecessary and wrong ... (more) |
— | almost 3 years ago |
| Edit | Post #282604 | Initial revision | — | almost 3 years ago |
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Why aren't the "21 possibilities here" NOT equally likely? Please see the last sentence below, that I highlighted in red. Example $1.4 .5$ (Ice cream cones). Suppose you are buying an ice cream cone. You can choose whether to have a cake cone or a waffle cone, and whether to have chocolate, vanilla, or strawberry as your flavor. This decision process ca... (more) |
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| Edit | Post #282603 |
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| Edit | Post #282603 | Initial revision | — | almost 3 years ago |
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Without calculations, how can you visualize "that half the squares are white and half are black"? 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 square now contain a black square, and vice versa". Example 1.4.4 (Chessboard). How many squares are t... (more) |
— | almost 3 years ago |
| Edit | Post #282602 |
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— | almost 3 years ago |
| Edit | Post #282602 | Initial revision | — | almost 3 years ago |
| Question | — |
"A occurred" vs. "something must happen" 1. Why doesn't "Something must happen" mean $s{actual} \in A$? 2. Scilicet, doesn't "A occurs" mean the same thing as "something must happen"? Something must happen. $\iff$ Some event must happen. $\iff$ At least one event must happen $\iff$ Call this event A. Then A occurred.  |
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| Edit | Post #282134 | Initial revision | — | almost 3 years ago |
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What story and TWO-digit Natural Numbers best fit Bayes' Theorem chart? Why did Madam Monica Cellio close What story and TWO-digit Natural Numbers best fit Bayes' Theorem chart? as duplicate of What story and ONE-digit Natural Numbers explain Bayes' Theorem chart most simply?? The difference is blindingly obvious. The first question seeks an example of Bayes' Theorem ... (more) |
— | almost 3 years ago |
| Edit | Post #281987 | Initial revision | — | almost 3 years ago |
| Question | — |
What story and two-digit Natural Numbers best fit Bayes' Theorem chart? To complete the table below most comfortably for teenagers, 1. what are the simplest stories? 2. what natural numbers $\le 99$ contrast the base rate fallacy the most? Please don't repeat a number. I'm trying to improve on this question that uses two-digits just $\le 20$, because 3. the... (more) |
— | almost 3 years ago |