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A. Juriˇ<strong>si</strong>ć in V. Batagelj: Verjetnostni račun in statistika 269 ✬ Iz Taylorjeve vrste ✫ e −x = 1 − x + x2 2! ocenimo 1 − x ≈ e −x in dobimo k−1 1 − i n i=1 Torej je verjetnost trčenja ≈ k−1 i=1 − x3 3! 1 − e −k(k−1) 2n . + · · · e −i n = e −k(k−1) 2n . Univerza v Ljubljani ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
A. Juriˇ<strong>si</strong>ć in V. Batagelj: Verjetnostni račun in statistika 270 ✬ Potem velja oziroma −k(k − 1) 2n in če ignoriramo −k, dobimo končno Za ε = 0, 5 je e −k(k−1) 2n ≈ 1 − ε V sploˇsnem je k proporcionalen s ✫ √ n. ≈ log(1 − ε), tj. k 2 − k ≈ 2n log k ≈ 2n log k ≈ 1, 17 √ n, 1 1 − ε . kar pomeni, da, če zgostimo nekaj več kot √ n elementov, je bolj verjetno, da pride do trčenja kot da ne pride do trčenja. Univerza v Ljubljani ▲ ❙ ▲ ▲ 1 1 − ε ▲ ● ❙ ▲ ▲ ☛ ✖ ✩ ✪
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✬ ✫ Ljubljana, 6. oktober 2008
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A. Juriˇsić in V. Batagelj: Verje
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