Prof. M.Vaisfeld
THE BINOMIAL PROBABILITY DISTRIBUTION
(see handout #6a)
6/8/2008
of the same basic experimental event. Each trial has only two possible outcomes
independent trials is
(#)
x = 3
P(x) =
0.3125
mean µ = np =
2.5
n = 5
P(X ≤ x) =
0.8125
1.25
p = 0.5
1 – P(X ≤ x) =
0.1875
standard deviation σ =
1.118034
Example of Linear Histogram of a binomial probability distribution.
10
0.5
x
P(X ≤ x)
0
1
0.0009765625
0.0009765625
1
10
0.009765625
0.0107421875
2
45
0.0439453125
0.0546875
3
120
0.1171875
0.171875
4
210
0.205078125
0.376953125
5
252
0.24609375
0.623046875
6
210
0.205078125
0.828125
7
120
0.1171875
0.9453125
8
45
0.0439453125
0.9892578125
9
10
0.009765625
0.9990234375
10
1
0.0009765625
1
obtained by means
of MS Excel function =COMBIN(n, x) .
BINOMIAL
vs
NORMAL
DISTRIBUTION
in the cells B51 and D51 (which display the copies of values stored in B29 and D29).
10
0.5
x
f(x) = NORMDIST()
5
0
1
0.0009765625
0.0017000733
1.5811388
1
10
0.009765625
0.0102848443
2
45
0.0439453125
0.0417071001
3
120
0.1171875
0.1133716522
4
210
0.205078125
0.206576619
5
252
0.24609375
0.2523132522
6
210
0.205078125
0.206576619
7
120
0.1171875
0.1133716522
8
45
0.0439453125
0.0417071001
9
10
0.009765625
0.0102848443
10
1
0.0009765625
0.0017000733
Binomial Probability Experiment
is an experiment that is made up of
repeated trials
(usually labeled as a success or a failure). There are
n
repeated
independent
trials,
so that the probability of success,
p
,
for each trial remains the same. The number
x
of
successful trials (the binomial random variable) may take on any
integer
value from
0
to
n
.
The
binomial probability
P(x)
of exactly
x
successes and
(n – x)
failures in
n
P(x)
=
n
C
x
p
x
(1 - p)
n–x
MS Excel has the function
=BINOMDIST(x, n, p, cumulative
) where the variables
x, n, p
have the sense explained above, and
cumulative
is a logical value
(
TRUE
or
FALSE
) that determines the form of the function. The function
BINOMDIST(x, n, p, FALSE
)
returns the probability
P(x)
(#) ;
the function
BINOMDIST(x, n, p, TRUE
) returns the
cumulative
distribution function
P(X ≤ x)
.
To evaluate the probabilities
P(x)
,
P(X ≤ x)
, the mean
µ
, and the variance
σ
2
type in
the values of
x,
n
,
and
p
in the cells
B23, B24
, and
B25
respectively.
variance σ
2
= np(1 – p) =
n
=
Probability
p
=
<== Vary the value of
0 ≤
p
≤ 1
in the cell D28
n
C
x
P(x)
=
n
C
x
p
x
(1 –
p
)
n–x
The values of combinations
n
C
x
in the 2
nd
column are
Vary the values of
n
and
p
in the cells B29 and D29 respectively,
not
below
n
=
Probability
p
=
n
C
x
P(x)
=
n
C
x
p
x
(1 –
p
)
n–x
µ = n*
p
=
σ = [n
p
(1 -
p
)]
1/2
=
A binomial distribution
with parameters
n
and
p
is approximately normal
for large
n
and
p
not too close to 1 or 0 [some books recommend
using this approximation only if
np
and
n(1 − p)
are both at least 5].
The approximating normal distribution has parameters