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1. Disregarding the messier LHS, how can I intuit the simpler RHS? I know [math]\color{green}{\dfrac{1E6}{T} = \Pr(}[/math] winning $1E6 in the Gold Ball Draw). 2. But why multiply this [math]\color{green}{\dfrac{1E6}T}[/math] by [math]\color{red}{\dfrac{69 - B}B}[/math]? 3. What does [math]\color{red}{\dfrac{69 - B}{B}}[/math] mean?

As per the link overhead, the jackpot increases by $2E6 for every ball removed, I know. But where did this(69−2B) stem from? I feel like the author, Paul Sinclair, pulled it out of a hat! The maximum Gold Ball Draw jackpot is $68E6, not $69E6. Then why did Paul Sinclair write 69?

What’s wrong with my reasoning in purple? Please pinpoint which sentence and step fails. I seek intuition. I DON’T want proofs, or formal arguments. Thanks! Presuppose lotteries let players pick any integer N≥32. Picking unpopular integers lowers your probability of winning lotteries. Why? Winning numbers range randomly from 1 to N. But you artificially narrow yourself to [32,N]. By disregarding [1,31], you flout the lottery’s random distribution of winning integers! As [1,N] contains more integers than [32,N], picking numbers ∈ [1,N] proffers more chances to win than picking ∈ [32,N]. Q.E.D.

My students still reckon Lande’s 2 motivations for ‘unless’ = ‘if not’ (below) too abstract, formalistic! Thus I need a 3rd simpler, motivation with merely etymology. But I never studied linguistics! How do I 1. use these etymology quotations, to motivate ‘unless’ = ‘if not’? 2. teach why “less than” means 'if not'? Traugott E.C. (1987) “UNLESS and BUT conditionals: a historical perspective.” In A. Athanasiadou and R. Dirven (eds.), On Conditionals Again. Amsterdam/Philadelphia: John Benjamins, page 145. Idem, page 156. Id. 157. See also Etymonline. Lande N.P. Classical logic and its rabbit-holes: A first course (2013), pages 55-7. Baronett S. Logic (5th edn 2022), 236 Gensler H. Introduction to Logic (3 edn 2017), 132.

While my hunky husband was lifting the chair up, and my brawny brother pushing the frame down, I kept hitting the top of the slide with a screwdriver, to try to loosen the slide. But these parts are still wedged too tightly! We ran out of energy and ideas! How can we life hack this please? https://imgur.com/Hu6rV57 GIF from this video. Illustration from this PDF. Chair manufacturer's page list these both.

Pls see screenshot below.

At https://www.scienceforums.net/topic/126544-why-did-i-lose-60-on-my-proof-of-generalized-vandermondes-identity-what-was-amiss/, 1. The tags antecede "Beneath is my proof graded 2/5". So why isn't it standing apart and alone? 2. Under ""Beneath is my proof graded 2/5", The 2nd quotation should stand alone and apart too.

My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof?

I grok, am NOT asking about, the answers below. Rather, how can I calculate the final answer DIRECTLY, without division? I don't know why my Latex isn't rendering here? Please see https://math.codidact.com/posts/285679. Orange underline 1. Unquestionably, $\color{#FFA500}{4 \times 3/2} = 3!$ But how can I construe 3! DIRECTLY WITHOUT DIVISION? What does 3! mean? Here's my surmisal. Blitzstein's solution hints to this calculation, but he didn't write 3! explicitly. You fix the first person. Then of the 3 people left, you can pick the 2nd person in your 2-person committee in 3 ways. Is this correct? Red underline 2. Unquestionably, ${\color{red}{\dfrac{8 \times 7 \times 6}{3!}}}$ = 8 × 7. But how can I construe 8 × 7 DIRECTLY WITHOUT DIVISION ? What does 8 × 7 mean? Here's my surmisal. You can pick the 1st committee member in 8 ways, and the 2nd member in 7 ways. But then can't you pick the 3rd member in 6 ways? By this Constructive Counting (p 38 bottom), the answer ought be 8 × 7 × 6??? Why isn't there 6? David Patrick, Introduction to Counting & Probability (2005), pp 66-7.

Two weeks ago, my carry on worked perfectly. But last week the detachable wheels stopped rotating smoothly, when I tried to glide my carry on. The wheels budge JUST A TAD when you push them hard, but it should be EFFORT LESS! Pls see two pictures below. I see nothing jammed in the wheels. I ventilated wheels with my hairdryer. I hosed water into wheels. But nothing fell out. Wheels are still stiff! Why? I bought this carry on for $350 USD. It is out of warranty. I can't afford buy new one.

English isn't my first language; I don't understand the difference between 'differential' and 'different' as adjectives. Why do these books use 'differential' as an adjective? Why not just write 'different'? 1. Stephen Darwall, Morality, Authority, and Law: Essays in Second-Personal Ethics I (2013), p 167. 2. Jeffrie G. Murphy, Philosophy Of Law: An Introduction To Jurisprudence (2018), Anyone know the page number? 3. The Routledge Companion to Philosophy of Law (2012), p 579. 4. Norman K. Denzin, Sociology of Law (2020) Anyone know the page number?

I saw this question at https://math.codidact.com/posts/283253.

My control variables are 1. water hardness level, as I live in the state of Florida, and I lodged at a Ritz Carlton in Florida. I'm not, in any way, affiliated with any corporation herein. 2. my eyeglasses with $700 USD Nikon SeeMax Ultimate lenses. 3. my cleaning method. 4. and my Nikon micro-fiber lens cleaning cloth. When I lodged at the Ritz Carlton hotel, I used their liquid hand wash to dab my eyeglasses. But even when I used different bottles of this hand wash, after rinsing with the hotel's tap water, my eyeglasses remain smudged. I didn't take the images below, but they resemble my smudges. When I got home, I daubed some of my Method Gel Hand Soap Refill, Sweet Water on my Nikon lenses as usual. Then rinsing with my home's tap water cleansed the smudges. Therefore Ritz Carlton's liquid hand wash appears to be the culprit for my smudges. But I never knew that some liquid hand washes can cause smudges?!!!?! What do you chemists think?

Are these books using 'differential' correctly? Why not just write 'different'? English isn't my first language, and I don't understand the difference between 'differential' and 'different' as adjectives. 1. Handbook of the Biology of Aging edited by Edward J. Masoro, Steven N. Austad. p 480. 2. The Cambridge Companion to the Philosophy of Biology edited by David L. Hull, Michael Ruse. p 46. 3. p 78. 4. Dictionary of Plant Genetics and Molecular Biology by Gurbachan Miglani. Anyone know the page #?

Thanks for your reply! I'm just wondering: Would sulfolane be a hypervalent compound? I had some trouble drawing its Lewis structure, especially for the electrons surrounding S.

Hello all: I am unsure if my answer to the following answer is correct. Could someone please confirm or refute it? Thank you! P.S.: If someone could tell me how to enter chemical formulas with LaTex without the italics, I would appreciate it as well! --- 1. What is the oxidation number of S in sulfolane [math]C_4 H_8 O_2 [/math] ? Structure: http://en.wikipedia.org/wiki/Image:Sulfolane-2D-skeletal.png --- Explanation for my work: My answer is +4 for S's oxidation number. According to the image, S is double bonded to the two oxygens so the formula must be [math]C_4 H_8 SO_2 [/math]. Therefore, I treated the [math]C_4 H_8 [/math] as a nonpolar hydrocarbon part so I assigned C and H oxidation numbers of -2 and +1 respectively. The entire molecule is neutral so I concluded that with [math]O_2[/math] being -4 in total (-2/O atom), S must be +4.

Hello everyone: Could someone please show me how to answer this question mathematically and correctly? I guessed the correct answer. Thank you. --- 1. When glucose -> CO2 + H2O during aerobic respiration, more than 60% of its energy is released as: a) oxygen b) carbon dioxide c) heat d) ATP e) NAD+ Correct answer: [hide]c) heat[/hide]