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Will the walker return?

Ukurasa huu bado haujatafsiriwa kwa Kiswahili, kwa hiyo unaonyeshwa kwa English.

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1

Swali

A walker tosses a fair coin every minute: heads, one step east; tails, one step west. Is she certain to come back to where she started? What if she can also step north and south — or up and down in space?

Fanya jaribio kabla ya kuendelea kusoma. Hakuna njia mbaya ya kuchunguza.

2

Jaribio

Maabara ya nasibu

Kila mtembeaji hupiga hatua kushoto au kulia kwa nasibu. Umbali wa kawaida kutoka mwanzo hukua kama √n.

-20020020406080100hatuamahali
  • matembezi 8 kati ya yote
  • umbali wa kipeuo cha wastani wa miraba (matembezi 200)
  • ±√n (kamili)

Umbali wa kipeuo cha wastani wa miraba baada ya hatua 100

Kwa uigaji10.94

Kamili√100 ≈ 10

Uigaji wa matembezi 200 ya hatua 100: umbali wa kipeuo cha wastani wa miraba 10.94. Thamani kamili ni √n, karibu 10.

Ushahidi, si uthibitishoHuu ni uigaji wa matembezi 200 (jaribio 1) kwa kutumia jenereta ya namba nasibu yenye mbegu. Ni ushahidi, si uthibitisho: thamani kamili zinatoka kwenye hisabati, na jaribio jipya hutoa namba tofauti kidogo.

Kwa usahihi: wastani wa (mahali)² baada ya hatua n ni n, kwa hiyo umbali wa kipeuo cha wastani wa miraba ni √n. Kufika mbali mara mbili kunahitaji hatua mara nne.

3

Vidokezo, ukivitaka

    4

    Kinachoendelea

    Kinachoendelea

    George Pólya answered this in 1921. On a line (one dimension) or a grid of streets (two dimensions), the walker returns home with probability 1 — in fact infinitely often. In three dimensions she does not: the chance of ever coming home is only about 0.340.34. A remark often attributed to Shizuo Kakutani puts it this way: a lost person will find their way home, but a lost bird may not.

    The reason is a sum. A walk returns home infinitely often exactly when the expected number of visits home is infinite, and that expected number is the sum, over all times, of the chance of being home then. After 2n2n steps the chance of being home behaves roughly like

    1πn  (on a line),1πn  (on a grid),a constant×n−3/2  (in space).\frac{1}{\sqrt{\pi n}} \;\text{(on a line)}, \qquad \frac{1}{\pi n} \;\text{(on a grid)}, \qquad \text{a constant} \times n^{-3/2} \;\text{(in space)}.

    The first two sums diverge — the second only just, like the harmonic series — so the walker comes home again and again. The third converges, so she may wander off for ever.

    There is a surprise even in one dimension: the walker is certain to return, yet the average time she takes to return is infinite. Random walks like these model the diffusion of molecules, stock prices, the spread of heat, and the search patterns of animals.

    Dokezo

    The simulation shows a finite number of walks for a finite time — evidence. Pólya's theorem is a proof about every walk forever.

    Imejaribiwa kikokotoziImeandaliwa na AI · inasubiri ukaguzi wa binadamu
    5

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