Galton box
Problem 14
Reporter: Krasikov Aleksey
Team: Voronezh One
Belgrade, Serbia 2015
Problem
In the Galton box, a regular 2D lattice
of obstacles disperses a thin flow of
falling particles. When falling on the
bottom of the box, the particles show a
normal distribution. Use various types
or particles and different arrangements
of the obstacles to find the conditions
when the distribution is no longer
normal
Investigation goals
1. Explain the occurrence the normal
distribution in Galton box
2. Find numerical border between
normal and binomial distribution
3. Identify the normal distribution
break parameter
4. Set an experiment and investigate
this phenomenon
What
is
a
Galton
box?
1123
Balls are thrown in the
center of the box
The obstacles are arranged
in a checkerboard pattern
Balls fall in the cell
at the bottom
4
How
does
it
work?
1123
The ball falls at the
first obstacle
Ball
Turn left
Turn right
First obstacle
1
𝑝=
2
1
𝑞=
2
It can bounce in the
right and left side
If conditions are
ideal, the probability
of a bounce in
different directions
will be equal
5
Check it
Modelling our experiments
in Algodoo
6
Check
it
1123
Modelling in Algodoo Physics
20
18
16
The number of balls
14
12
10
8
6
4
2
0
Columns
7
How
does
it
work?
1123
Turn left (1/2)
The ball is involved in
two independent trials
Turn right (1/2)
First series
Turn left (1/2)
Turn right (1/2)
Second series
In each such test ball
can bounce to the
right side or to the left
side
Second series
1
𝑝=
4
1
𝑠=
2
1
𝑞=
4
Binomial distribution
8
Check
it
1123
Modelling in Algodoo Physics
20
18
16
The number of balls
14
12
10
8
6
4
2
0
Columns
9
How does it work?
First series
Second series
If 𝑛 → ∞, the distribution
tends to normal
Second series
...
N series
10
Binomial vs normal distribution
The binomial distribution is a
distribution of the numbers of
"successes" in n independent
experiments (probability is
constant)
11
Binomial vs normal distribution
The binomial distribution is
discrete, because it is
concentrated at several points.
E.g.: The number of heads (or tails)
has binomial distribution
12
Binomial vs normal distribution
The normal distribution is
continuous. It coincides
with the Gaussian function
13
Normal distribution
Binomial distribution turns to normal for certain
values of the parameters. E.g.: This occurs when
the Galton box has a large number of series with
obstacles (n>10)
Binomial distribution → Normal distribution
14
Check it
n=1
n=3
n=6
40
30
50
35
30
20
30
The number of balls
The number of balls
The number of balls
25
40
25
20
15
20
15
10
10
10
5
5
0
0
Columns
0
Columns
Columns
Modelling in “Galton box”
Binomial distribution → Normal distribution
15
Check it
n=12
n=9
45
30
40
35
The number of balls
The number of balls
25
20
15
10
30
25
20
15
10
5
5
0
0
Columns
Columns
Modelling in “Galton box”
Binomial distribution → Normal distribution
16
When does Galton box work?
R
−1𝛿 −𝜇
𝜇 - expected value
𝜇
1𝛿
𝛿 - standard deviation
Modelling in Algodoo. We
can see normal distribution
If 𝛿 ≥ 𝑅, Galton
box works
17
When doesn’t Galton box work?
R
Modelling in Algodoo.
We don’t see normal
distribution
If 𝛿 < 𝑅, Galton box
doesn’t work
−1𝛿 −𝜇
𝜇 - expected value
𝛿 - standard deviation
𝜇 1𝛿
18
Experimental investigation
𝑓(𝑥) =
1
𝛿 2𝜋
(𝑥−𝜇)2
−
𝑒 2𝛿2
Gaussian function:
𝛿- standard deviation
𝜇- expected value
19
Experimental investigation
−𝑡
𝛿=
𝑖=−1
−𝑥2
−𝑥1
−1
𝑘𝑖 2
𝑥𝑖 +
𝑘
0
1
𝑡
𝑖=1
𝑥1
𝑘𝑖 2
𝑥𝑖
𝑘
𝛿- standard deviation
2𝑡 + 2- number of columns
𝑥𝑖 - coordinate of i-column
𝑘𝑖 - the number of balls in
the i-column
𝑘𝑖 - the total number of balls
𝑥2
20
Experimental investigation
Variable parameters:
1. the ratio between the distance from one
pin to another and the radius of the ball;
2. the shape of particles;
3. the shape of obstacles
If 𝛿 < 𝑅, Galton
box doesn’t work
Find out the 𝛿
If 𝛿 ≥ 𝑅, plot the
Gaussian functions
Experiments
21
1) Ratio between distance and radius
𝐿
𝑆=
𝑑
Ball
First series
d
Second series
Second series
L
S=1
S=6
S=20
22
1) Ratio between distance and radius
S=1
R=1
−1
𝛿 = 1,54
0
1
23
1) Ratio between distance and radius
S=1
practice line
theoretical line
0.3
𝛿>𝑅
1,54 > 1
Probability
0.25
0.2
0.15
0.1
0.05
0
-5
-4
-3
-2
-1
0
Column
1
2
3
4
5
There for we can see, that the Galton box works
24
1) Ratio between distance and radius
S=6
S=20
R
R
𝛿 < 𝑅, that’s why Galton box doesn’t work
25
Real experiment setup
L=5 mm
Changing the frequency of obstacles
L=10 mm
26
1) Ratio between distance and radius
𝛿>𝑅
1,75 > 1
𝛿>𝑅
1,43 > 1
theoretical line
theoretical line
practise
0.3
0.25
0.25
0.2
Probability
Probability
0.2
0.15
0.1
0.15
0.1
0.05
0.05
0
0
-6
-4
-2
0
Columns
practise
2
4
6
-6
-4
-2
0
Columns
2
4
6
27
1) Ratio between distance and radius
𝛿≈𝑅
𝛿≈1
𝛿<𝑅
𝛿 = 0,85
Galton box doesn’t work
28
Explanation
Small particles pass
through without collision,
so particles concentrate
in the central column
First series
Second series
Second series
29
2) Shape of particles
Oval
Triangle
Square
Star
Triangle with a displaced
center of mass
30
2) Shape of particles
Modelling in Algodoo Physics
31
2) Shape of particles
Modelling in Algodoo Physics
32
Explanation
Despite the fact that we change the shape of the
particles, the possibility of deviation to the right
and left side persists.
That’s why we can see the normal distribution
33
3) Shape of obstacles
Obstacles in the shape
of triangles
34
3) Shape of obstacles
Explanation: we change the shape of obstacles, so the probability of
deviation to the left side become bigger. Therefore, the normal distribution
was broke
35
Conclusion
1. We explained how Galton box
works
2. Modelled and made an experiment
in reality
3. Explained difference between
normal and binomial distribution
4. If the number of series strive to
infinity, the distribution tends to
normal
Conclusion
5. If standard deviation smaller then
distance between columns, Galton
box doesn’t work, in another way it
works. If particles is very small,
normal distribution will be break
6. Distribution doesn’t depends on the
shape of particles (and center of
mass of the one particle)
7. If we change the probability of
deviation (change shape of
obstacles), normal distribution will
be break
Thanks for attention!
38