Timeline for Which line is more likely to intersect the circle?
Current License: CC BY-SA 4.0
28 events
| when toggle format | what | by | license | comment | |
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| Sep 14 at 2:21 | comment | added | Dan | @Tyo There is already a link to the cross-post at the end of the question :) | |
| Sep 14 at 2:14 | comment | added | Tyo | Cross posted here. math.stackexchange.com/questions/5089940/… | |
| Aug 18 at 2:44 | answer | added | Theguyabovemeislying | timeline score: 0 | |
| Aug 14 at 4:35 | history | edited | Dan | CC BY-SA 4.0 | added 94 characters in body |
| Aug 12 at 6:34 | answer | added | Kilian | timeline score: 2 | |
| Aug 11 at 1:57 | history | edited | Dan | CC BY-SA 4.0 | added 42 characters in body |
| Aug 11 at 0:16 | comment | added | Dan | @Lucenaposition The vertices of a pentagram are random points on a circle. Let $a,b,c$ be the areas of three consecutive triangular "petals". Which is more likely to be larger: $a^2$ or $bc$? Simulations suggest that $bc$ is larger approximately $50.005\%$ of the time. (I don't think it's exactly $1/2$.) | |
| Aug 10 at 23:14 | comment | added | Lucenaposition | @Dan You could also post some where they were not exactly equal but very close to equal. These could also be interesting. | |
| Aug 10 at 23:12 | comment | added | Dan | @Lucenaposition Yes, I knew by integration. First, I did simulations that suggested they were equally likely. Then I worked out the integrals. Then I posted the question. | |
| Aug 10 at 23:10 | comment | added | Lucenaposition | @Dan you knew by integration they were equally likely, right? | |
| Aug 10 at 23:10 | comment | added | Dan | @Lucenaposition I knew they were equally likely before I posted the question. | |
| Aug 10 at 23:08 | comment | added | Lucenaposition | @Dan did you know they were equally likely when you posted the question, or did you only find out later? | |
| Aug 10 at 17:12 | answer | added | Dan | timeline score: 9 | |
| Aug 10 at 16:32 | comment | added | Benjamin Wang | @Dan maybe this question should also be on Math StackExchange. | |
| Aug 10 at 10:11 | comment | added | Lucenaposition | @Dan Post the integration answer as an answer. | |
| Aug 10 at 6:28 | comment | added | Pranay | Ah I see. I found some different integral expressions for the probabilities and Mathematica gave me their exact values in terms of polylog function, which is also what you see. The numerical value you quote agrees with what I got. | |
| Aug 10 at 6:20 | comment | added | Dan | @Pranay rot13 V nyfb hfrq vagrtengvba gb fubj gung gur cebonovyvgvrf ner gur fnzr, ohg V fhfcrpg gurer vf n zber ryrtnag zrgubq. Fb vg'f n chmmyr sbe zr, gbb. Ol gur jnl, obgu cebonovyvgvrf ner $\frac{2}{\pi^2}\int_0^\pi\operatorname{arccot}\left(2\sqrt2\sin x\right)dx=\frac{2}{\pi^2}\left(\frac38\left(\pi^2-\ln^22\right)-\operatorname{Li}_2\left(-\sqrt{2}\right)-3\operatorname{Li}_2\left(\frac{1}{\sqrt{2}}\right)\right)=0.3871287106\dots$ | |
| Aug 10 at 5:28 | comment | added | Pranay | @Lucenaposition. My answers also agree with yours up to the first three decimals. | |
| Aug 10 at 5:22 | comment | added | Pranay | @Lucenaposition. I see. I found integral expressions for both the probabilities (it’s much simpler for BC than AB) and then computed them numerically. They are equal to whatever decimal places I chose. So I believe they are the same. I have some ideas on showing the integrals are the same but it feels far too complicated and not what Dan intended. | |
| Aug 10 at 5:20 | comment | added | Lucenaposition | It was, and I hoped nothing went wrong. | |
| Aug 10 at 5:18 | comment | added | Pranay | @Lucenaposition. Was this result obtaining by picking points on the circles (pseudo)randomly? | |
| Aug 8 at 17:35 | vote | accept | Dan | ||
| Aug 8 at 21:33 | |||||
| Aug 8 at 15:52 | answer | added | Nuclear Hoagie | timeline score: -3 | |
| Aug 8 at 9:25 | history | became hot network question | |||
| Aug 8 at 1:33 | history | edited | Lucenaposition | CC BY-SA 4.0 | added 3 characters in body |
| Aug 8 at 1:05 | comment | added | Lucenaposition | [Not an answer]: If my code works, I think the probability of intersection is the same (or very similar) and around 0.387. | |
| Aug 7 at 23:04 | history | edited | bobble♦ | CC BY-SA 4.0 | removed unnecessary MathJax https://puzzling.meta.stackexchange.com/questions/7421/a-brief-guide-to-mathjax-for-pse-users |
| Aug 7 at 22:55 | history | asked | Dan | CC BY-SA 4.0 |