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We have a square. we will opt three random point from inside of this square and name it $p_{1},p_{2},p_{3}$ then opt another random point $p_{4}$. what is the probability of that $p_{4}$ lies in triangle $p_{1}p_{2}p_{3}$ ?

I write program and test it 10 times and in each case test it for $10^{7}$ points and the results are:

TEST CASE # 1 = 0.076566
TEST CASE # 2 = 0.076430
TEST CASE # 3 = 0.076308
TEST CASE # 4 = 0.076378
TEST CASE # 5 = 0.076433
TEST CASE # 6 = 0.076340
TEST CASE # 7 = 0.076289
TEST CASE # 8 = 0.076382
TEST CASE # 9 = 0.076402
TEST CASE #10 = 0.076260
Karo
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  • I believe this is sufficient to answer your question. http://people.missouristate.edu/lesreid/Adv41.html

    Once you have the expected area, the probability $p_{4}$ lies in your triangle should follow.

     – JessicaK Jun 01 '14 at 09:23
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    @JessicaK. This looks like magics to me ! Thanks for the link. – Claude Leibovici Jun 01 '14 at 09:27
  • @JessicaK: Link not available, are there some regional restrictions on that site? Other link to and partial answer at http://www.rqna.net/qna/pkzyz-expected-area-of-a-triangle-determined-by-randomly-placed-points.html – Lutz Lehmann Jun 01 '14 at 11:57
  • Link's not working for me either. Looks like a related question was asked before, which has that link. – gar Jun 01 '14 at 12:04
  • Are you looking for an exact answer, in a formula, or just an approximation? What did the original question ask for, or was it just curiosity? – Henno Brandsma Jun 01 '14 at 13:01
  • @LutzL The link works fine for me, I do not know why it would not work. – JessicaK Jun 06 '14 at 04:55

1 Answers1

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A mathworld link and a paper answers the question.

$$ \mathbb{P} = \frac{11}{144} \approx 0.0763889 $$

gar
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