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| another puzzle; responding to my critics | |
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| Tweet Topic Started: Jul 14 2016, 06:09 AM (272 Views) | |
| jon-nyc | Jul 14 2016, 06:09 AM Post #1 |
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Cheers
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Ok, this one is new, I'm making it up on the fly. Consider a dart board of radius R. You throw two darts at it. Each throw hits the board at a random spot. Both throws are independent. All points on board have equal probability of being hit. What is the probability that the two darts will hit within R/4 of each other? Assume perfect circle board, infinitesimal dart tip, etc. (If you brute force a solution with code, somewhere a kitten cries after failing his SAT test) |
| In my defense, I was left unsupervised. | |
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| Klaus | Jul 14 2016, 07:02 AM Post #2 |
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HOLY CARP!!!
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Interesting. I predict that the solution contains some weird integral. But could you clarify what you mean by "All points on board have equal probability of being hit"? The probability of hitting a concrete point is 0. I guess what you mean here is this: If A is a subset of the unit circle, then the probability of hitting a point in A is P(A) = |A| / 3.1416, where |A| is the size/area of A. (in maths geek speech, one would also need to assume that A is a member of the Borel σ-algebra of the unit circle). |
| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| Klaus | Jul 14 2016, 08:25 AM Post #3 |
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HOLY CARP!!!
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Spoiler: click to toggle
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| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 14 2016, 09:53 AM Post #4 |
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Cheers
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Yes on the probability clarification. And show your work! |
| In my defense, I was left unsupervised. | |
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| Klaus | Jul 14 2016, 01:06 PM Post #5 |
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HOLY CARP!!!
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Spoiler: click to toggle
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| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 14 2016, 01:35 PM Post #6 |
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Cheers
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Spoiler: click to toggle
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| In my defense, I was left unsupervised. | |
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| George K | Jul 14 2016, 01:46 PM Post #7 |
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Finally
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I love it when you guys talk dirty. |
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A guide to GKSR: Click "Now look here, you Baltic gas passer... " - Mik, 6/14/08 Nothing is as effective as homeopathy. I'd rather listen to an hour of Abba than an hour of The Beatles. - Klaus, 4/29/18 | |
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| Klaus | Jul 14 2016, 02:07 PM Post #8 |
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HOLY CARP!!!
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Spoiler: click to toggle
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| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 14 2016, 02:30 PM Post #9 |
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Cheers
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Does this help? Spoiler: click to toggle
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| In my defense, I was left unsupervised. | |
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| Klaus | Jul 14 2016, 02:47 PM Post #10 |
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HOLY CARP!!!
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How does the step from (4) to (5) work? Looks like magic to me. |
| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 14 2016, 03:22 PM Post #11 |
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Cheers
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It does seem pretty slick. Like James Bond came in and solved your calculus problem for you. |
| In my defense, I was left unsupervised. | |
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| Klaus | Jul 14 2016, 11:52 PM Post #12 |
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HOLY CARP!!!
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Show us the solution then, Jon! |
| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| Klaus | Jul 15 2016, 12:10 AM Post #13 |
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HOLY CARP!!!
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By the way, Jon, maybe you'll find it interesting that there are dedicated programming languages to solve these kinds of problems, "probabilistic programming languages". Just for fun, here's a model of your problem in Church. I've asked it to display a histogram of the distances in addition to computing the percentage of throws with distance smaller than R/4. ![]() If you ignore the funny LISP-style prefix syntax, you can see that it's rather straightforward. I just draw Cartesian coordinates uniformly, ask for their distance (line starting with "sqrt") but then condition the model to only consider coordinates in the unit circle (line starting with "and"). If you'd like to play around with this, you can try it in your browser here. This particular example doesn't show-case the real strength of these kinds of languages, namely if you have a very high-dimensional problem space (many random variables) and a very low acceptance rate (i.e. the conditioning rejects, say, 99.999999% of all runs - in your example only 100/pi percent are rejected). Then you need sophisticated sampling algorithms. In this case I used a so-called "Metropolis-Hastings" algorithm - that's what "mh-query" means. |
| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 15 2016, 12:48 AM Post #14 |
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Cheers
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I haven't solved it yet! I made it up on the fly, remember. |
| In my defense, I was left unsupervised. | |
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| Klaus | Jul 15 2016, 01:55 AM Post #15 |
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HOLY CARP!!!
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Ah, so the real challenge is that there may not even be an analytic solution, or that an analytic solution requires to solve some Fields-medalesque problem!
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| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 15 2016, 02:05 AM Post #16 |
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Cheers
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Did you see the probability function at the bottom of my link? You just integrate that from s= 0->0.25 and you'll get the answer. But I haven't yet figured out how they derived that function. |
| In my defense, I was left unsupervised. | |
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| Klaus | Jul 15 2016, 02:08 AM Post #17 |
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HOLY CARP!!!
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(by the way, I get the same result for "disk line picking" than in your link, see updated screenshot) |
| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 15 2016, 02:10 AM Post #18 |
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Cheers
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Nice. |
| In my defense, I was left unsupervised. | |
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| Klaus | Jul 15 2016, 02:12 AM Post #19 |
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HOLY CARP!!!
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In addition to integration, you'd also need to divide the result by some constant, I guess. The book where they derive that formula is just $69. I think you owe it to the community here to buy that book and show us how to derive the formula
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| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 15 2016, 02:16 AM Post #20 |
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Cheers
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I don't think a constant is needed. |
| In my defense, I was left unsupervised. | |
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| Klaus | Jul 15 2016, 02:19 AM Post #21 |
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HOLY CARP!!!
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The integral on that web page does seem to have an analytic solution.![]() And the result agrees with my numerical solution: 0.055. Heureka!? |
| Trifonov Fleisher Klaus Sokolov Zimmerman | |
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| jon-nyc | Jul 15 2016, 04:01 AM Post #22 |
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Cheers
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Let's call this puppy solved! You can see his derivation on pp 128-129 of his book on Google books. |
| In my defense, I was left unsupervised. | |
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