An
example
on
binomial
probabili.es,
inspired
by
the
book
The
Science
of
Fear
by
Daniel
Gardner
ECE
302
Fall
2009
TR
3‐4:15pm
Purdue
University,
School
of
ECE
Prof.
Ilya
Pollak
Binomial
r.v.’s
and
journalism
majors
•  Excerpts
from
The
Science
of
Fear
by
D.
Gardner
(DuSon,
2008).
–  The
probability
of
the
earth
being
walloped
by
a
300‐meter
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
the
century.
Binomial
r.v.’s
and
journalism
majors
•  Excerpts
from
The
Science
of
Fear
by
D.
Gardner
(DuSon,
2008).
–  The
probability
of
the
earth
being
walloped
by
a
300‐meter
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
the
century.
…
For
a
100‐meter
rock,
the
odds
are
1
in
10,000
in
one
year
and
1
in
100
over
the
next
100
years.
Binomial
r.v.’s
and
journalism
majors
•  Excerpts
from
The
Science
of
Fear
by
D.
Gardner
(DuSon,
2008).
–  The
probability
of
the
earth
being
walloped
by
a
300‐meter
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
the
century.
…
For
a
100‐meter
rock,
the
odds
are
1
in
10,000
in
one
year
and
1
in
100
over
the
next
100
years.
At
30
meters,
the
odds
are
1
in
250
per
year
and
1
in
2.5
over
the
next
100
years.
Binomial
r.v.’s
and
journalism
majors
•  Excerpts
from
The
Science
of
Fear
by
D.
Gardner
(DuSon,
2008).
–  The
probability
of
the
earth
being
walloped
by
a
300‐meter
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
the
century.
…
For
a
100‐meter
rock,
the
odds
are
1
in
10,000
in
one
year
and
1
in
100
over
the
next
100
years.
At
30
meters,
the
odds
are
1
in
250
per
year
and
1
in
2.5
over
the
next
100
years.
•  Excerpt
from
an
online
review
of
the
book:
–  He
says
'The
probability
of
the
earth
being
walloped
by
a
300‐metre
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
a
century.'
No
it
doesn't.
Binomial
r.v.’s
and
journalism
majors
•  Excerpts
from
The
Science
of
Fear
by
D.
Gardner
(DuSon,
2008).
–  The
probability
of
the
earth
being
walloped
by
a
300‐meter
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
the
century.
…
For
a
100‐meter
rock,
the
odds
are
1
in
10,000
in
one
year
and
1
in
100
over
the
next
100
years.
At
30
meters,
the
odds
are
1
in
250
per
year
and
1
in
2.5
over
the
next
100
years.
•  Excerpt
from
an
online
review
of
the
book:
–  He
says
'The
probability
of
the
earth
being
walloped
by
a
300‐metre
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
a
century.'
No
it
doesn't.
That's
like
saying
'The
odds
of
ge\ng
a
head
with
one
throw
of
a
coin
is
1
in
2,
which
makes
the
odds
1
in
1
over
two
throws.'
That's
not
how
probabili.es
combine.
Binomial
r.v.’s
and
journalism
majors
•  Excerpts
from
The
Science
of
Fear
by
D.
Gardner
(DuSon,
2008).
–  The
probability
of
the
earth
being
walloped
by
a
300‐meter
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
the
century.
…
For
a
100‐meter
rock,
the
odds
are
1
in
10,000
in
one
year
and
1
in
100
over
the
next
100
years.
At
30
meters,
the
odds
are
1
in
250
per
year
and
1
in
2.5
over
the
next
100
years.
•  Excerpt
from
an
online
review
of
the
book:
–  He
says
'The
probability
of
the
earth
being
walloped
by
a
300‐metre
asteroid
in
any
given
year
is
1
in
50,000,
which
makes
the
odds
1
in
500
over
the
course
of
a
century.'
No
it
doesn't.
That's
like
saying
'The
odds
of
ge\ng
a
head
with
one
throw
of
a
coin
is
1
in
2,
which
makes
the
odds
1
in
1
over
two
throws.'
That's
not
how
probabili.es
combine.
•  Interes.ngly,
they
are
both
wrong.
Some
assump.ons
•  Assume
that
Gardner’s
sta.s.cs
of
asteroid
hits
for
one
year
are
correct:
1/50000
probability
for
a
300‐meter
asteroid,
1/10000
for
a
100‐meter,
and
1/250
for
a
30‐
meter.
Some
assump.ons
•  Assume
that
Gardner’s
sta.s.cs
of
asteroid
hits
for
one
year
are
correct:
1/50000
probability
for
a
300‐meter
asteroid,
1/10000
for
a
100‐meter,
and
1/250
for
a
30‐
meter.
•  Assume
that
asteroid
hits
in
different
years
are
independent,
and
that
there
cannot
be
more
than
one
hit
per
year.
Some
assump.ons
•  Assume
that
Gardner’s
sta.s.cs
of
asteroid
hits
for
one
year
are
correct:
1/50000
probability
for
a
300‐meter
asteroid,
1/10000
for
a
100‐meter,
and
1/250
for
a
30‐
meter.
•  Assume
that
asteroid
hits
in
different
years
are
independent,
and
that
there
cannot
be
more
than
one
hit
per
year.
•  If
P(1
hit
in
1
year)
=
p
then
P(≥1
hits
in
100
years)
=
1
–
P(0
hits
in
100
years)
=
1
–
(1
–
p)
100
Call
this
quan.ty
f(p).
Some
calcula.ons
•  f(p)
=
P(≥1
hits
in
100
years)
=
1
–
(1
–
p)
100
Taylor series :
f ( p) = f (0) + f '(0) p +
f ''(0) 2
p +…
2!
Some
calcula.ons
•  f(p)
=
P(≥1
hits
in
100
years)
=
1
–
(1
–
p)
100
Taylor series :
f ''(0) 2
f ( p) = f (0) + f '(0) p +
p +…
2!
When p is very small,
f ( p) ≈ f (0) + f '(0) p = 100 p (because f '( p) = 100(1− p) 99 )
Some
calcula.ons
•  f(p)
=
P(≥1
hits
in
100
years)
=
1
–
(1
–
p)
100
Taylor series :
f ''(0) 2
f ( p) = f (0) + f '(0) p +
p +…
2!
When p is very small,
f ( p) ≈ f (0) + f '(0) p = 100 p (because f '( p) = 100(1− p) 99 )
E.g., if p = 1/50000, then f ( p) ≈ 100 /50000 = 1/500 = 0.002 (actually, 0.001998)
Some
calcula.ons
•  f(p)
=
P(≥1
hits
in
100
years)
=
1
–
(1
–
p)
100
Taylor series :
f ''(0) 2
f ( p) = f (0) + f '(0) p +
p +…
2!
When p is very small,
f ( p) ≈ f (0) + f '(0) p = 100 p (because f '( p) = 100(1− p) 99 )
E.g., if p = 1/50000, then f ( p) ≈ 100 /50000 = 1/500 = 0.002 (actually, 0.001998)
E.g., if p = 1/10000, then f ( p) ≈ 100 /10000 = 1/100 = 0.01 (actually, 0.009951)
Some
calcula.ons
•  f(p)
=
P(≥1
hits
in
100
years)
=
1
–
(1
–
p)
100
Taylor series :
f ( p) = f (0) + f '(0) p +
f ''(0) 2
p +…
2!
When p is very small,
f ( p) ≈ f (0) + f '(0) p = 100 p (because f '( p) = 100(1− p) 99 )
E.g., if p = 1/50000, then f ( p) ≈ 100 /50000 = 1/500 = 0.002 (actually, 0.001998)
E.g., if p = 1/10000, then f ( p) ≈ 100 /10000 = 1/100 = 0.01 (actually, 0.009951)
But when p = 1/250, the quadratic term in the Taylor series is no longer negligible :
f ''(0) 2
9900
p =−
= −0.0792 (because f ''( p) = −9900(1− p) 98 )
2!
2 ⋅ 62500
Some
calcula.ons
•  f(p)
=
P(≥1
hits
in
100
years)
=
1
–
(1
–
p)
100
Taylor series :
f ( p) = f (0) + f '(0) p +
f ''(0) 2
p +…
2!
When p is very small,
f ( p) ≈ f (0) + f '(0) p = 100 p (because f '( p) = 100(1− p) 99 )
E.g., if p = 1/50000, then f ( p) ≈ 100 /50000 = 1/500 = 0.002 (actually, 0.001998)
E.g., if p = 1/10000, then f ( p) ≈ 100 /10000 = 1/100 = 0.01 (actually, 0.009951)
But when p = 1/250, the quadratic term in the Taylor series is no longer negligible :
f ''(0) 2
9900
p =−
= −0.0792 (because f ''( p) = −9900(1− p) 98 )
2!
2 ⋅ 62500
f (0) + f '(0) p = 100 /250 = 0.4,
whereas f ( p) ≈ 0.330217