# Fluctuation Première S

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So in Mathematics at 1st level S <br>
we're not supposed to talk about the notion <br>
independence in probability
but it is not forbidden to educate <br>
students when they speak of the law <br>
binomial.
Especially if the binomial distribution is introduced <br>
with classical Galton board which <br>
is already useful to talk
concretely coefficients <br>
binomial in 1st year level S
before extending in the Terminal S <br>
Moivre-Laplace theorem.
In fact most devices <br>
this experimental Galton board, <br>
whether real or simulated,
pay little attention to two assumptions <br>
independence:
the first independent <br>
successive bifurcations of a ball in <br>
his fall.
and hypothesis 2:
if we ever launched simultaneously
a large number of balls,
independence between the beads.
2 simulations with 2 sizes of beads <br>
show different
phenomena bottling and Filmography
workaround we have just seen,
and, conversely, with balls more <br>
small, a phenomenon of liquefaction and Filmography
drive
which produces an abnormality vis-à-vis this <br>
as predicted by theory,
here you will get carried away by the momentum of <br>
fluid is going to say,
two vertices on the ends rather <br>
at the center.
So to really talk about law <br>
binomial with a Galton board and Filmography
So seriously talk of independence,
successive rebounds on rows <br>
nails, it is welcome that make a <br>
simulation.
So what it gives here? If I <br>
not going too fast,
first drop gives 3 rebounds <br>
a second straight 4 gives a <br>
third gives 7
and therefore as and when it appears as <br>
in the table in the diagram <br>
right in Figure 2,
experimental frequencies <br>
successive
should gradually
closer asymptotically
our theoretical probabilities.
I get tired of it, so I'll <br>
increase speed,
I reset and then rather do <br>
fall
100
100
logs, although of course we do this <br>
plus chuttes,
100 by 100 and then maybe support 20 <br>
time for 2000
2000 beads falling ...
Obtain frequencies and therefore
beginning to adjust to the <br>
theoretical probabilities.
So it allows us to speak of <br>
intervals fluctuation which <br>
way? As follows ...
I open a new window. <br>
Then under the Insert Text tool, there is a <br>
interesting module Probability Calculations
we are the first S so we <br>
speak as the binomial distribution,
Here we had earlier 10 rows. <br>
Then we will arrange a little more room <br>
for our business.
What is very interesting is that we <br>
can speak of the distribution function <br>
like that, with the "left" tab,
simply a value from 0 to I <br>
gives in this field, say 7 or with <br>
the tab is here
And so the question for <br>
determine the range of fluctuation in <br>
confidence level 95%
half the risk of saying something stupid that <br>
would be 2.5%
underperformance and presto, that we <br>
immediately gives k1 = 2
See that it took the value <br>
immediately above 2.5%, or <br>
probability
the next higher value <br>
0,025
And on the other side, then, for <br>
outperformance could say,
you hit .975 in this field and it will <br>
fit the value immediately <br>
top,
and a k2
k1 = 2 and k2 = 8
So obviously, with very little <br>
rows of nails, it is not very <br>
impressive, but
the processing speed is very precious <br>
when compared to that of calculators
There, like, I want to know if I put k1 <br>
0025 and I get 40,
and if I want to know k2, I put 0975,
and I get k2 = 60. So obviously,
with probability p = 1/2, there is a <br>
symmetry but if the symmetry is broken
putting, for example, p = pi / 4, <br>
same case,
k1 is equal to 70 <br>
and Filmography
k2 is equal to 86.
Then the a priori risk has no reason <br>
to be set once for all 5%
this can be a risk in various <br>
concrete situations
largest to smallest, it would give <br>
various fluctuation intervals,
but in the first program S
one remains in this interval fluctuation <br>
5%.