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4 Monte Carlo methods

domnumbers differed, i.e. the pebbles landed at different locations in
each run.
Weshall return later to this table when computing the statistical er-
rors to be expected from Monte Carlo calculations.In the meantime, we
intend to show that the Monte Carlo methodis a powerful approach for
the calculation ofintegrals (in mathematics, physics, and other fields).
But let usnot get carried away:none ofthe results in Table 1.1 has
fallen within the tight errorbounds already knownsince Archimedes
fromcomparing a circle with regular n-gons:

10 1
3.141 3 < < 3 3.143. (1.1)
71 7
The children’s valuefor is very approximate, butimproves and finally
becomes exact in the limit of an infinite number oftrials.This is Jacob
Bernoulli’s weak lawof large numbers (see Subsection 1.3.2). The chil-
dren also adopt a very sensiblerule: they decide onthe total number of
throwsbefore starting the game.The other day, in a game of “N=4000”,
they had at some point 355 hits for452trials—this gives a very nice ap-
proximationto the book value of .Withouthesitation, theywent on
355 355 1
= = × 3.14159292 .. .
452 4 × 113 4 until the 4000th pebble was cast.Theyunderstand that one must not
/4= 1 × 3.14159265 .. . stopa stochastic calculationsimply because the resultis just right, nor
4
should one continueto play because the resultis notclose enough to
what we think the answer shouldbe.
1.1.2 Markov-chain sampling

In Monte Carlo,it isnot only children who play at pebble games.We
can imagine that adults, too,may play their own versionat the local
heliport, in the late evenings.After stowing away all their helicopters,
theywander around the square-shaped landing pad (Fig.1.2), which
looks just like the area in the children’s game, only bigger.

Fig. 1.2 Adults computing the number at the Monte Carlo heliport.