Transcript Slide 1

The Galton board
The Galton board (or
Quincunx) was devised by
Sir Francis Galton to
physically demonstrate
the relationship between
the binomial and normal
distributions.
The Galton box consists of pegs arranged in a
triangular pattern.
Balls (or beans) are dropped from the top of the
board, bounce among the pegs, and collect in
bins at the bottom.
At each peg the ball can fall left or right.
If we consider a left turn to be a FAILURE and
a right turn to be a SUCCESS we can see each
pin as a Bernoulli experiment.
FAILURE
SUCCESS
Each row in the Galton board represents an
independent Bernoulli trial.
The board below represents a Binomial
experiment with 3 trials.
Trial 1
Trial 2
Trial 3
0
1
2
3
The bins represent the
number of times that
the ball moves to the
left (failure) or right
(success) at each level.
The number of times we get 0 right
turns (0 “successes”)
The number of times we get 1 right
turns (1 “successes”)
The number of times we get 2 right
turns (2 “successes”)
The number of times we get 3 right
turns (3 “successes”)
0
1
2
3
Suppose that the probability that a ball moves
to the right when it hits a peg is p=0.5.
The paths that the ball
can follow to land in bin
“1” are:
{R,L,L}, {L,L,R}, or {L,R,L}.
i.e., 3 unique paths.
The probability of one of
these paths is equal to:
p ⨯ (1-p)⨯ (1-p) =0.125.
The probability
that
R
the ball landsLin bin
“1” is then:L
3 ⨯ 0.125=0.375
0
1
2
3
The probability that a ball lands in bin x is
thus given by the binomial mass function
(with n trials and probability of success equal
to p):
The combination term
represents the
number of unique paths that the ball can
follow.
Pascal’s Triangle
Pascal’s triangle
can also be used to
determine the
number of unique
paths that the ball
can follow.
This is the number
of unique paths for
each bin in our
example.
What is the relationship
between the Normal
distribution and the
Binomial distribution?
A well known property is that if the number of
trials increases to infinity, then the Binomial
distribution approximates the Normal
distribution.
This approximation is reasonable even for a small
number of trials.
The Normal distribution result is due
to the Central Limit Theorem.
The Central Limit Theorem states that the sum
of n independent identically distributed random
variables is approximately Normally distributed
as n becomes large.
The following animation
can illustrate this
approximation for various
numbers of trials.
Note the shape of the frequency bar plot
constructed at the bottom of the animation.
Fin