10
0
1
2
10 1
1024 0.1%
×=≈
10
1
1
2
10 10
1024 1.0%
×=≈
10
2
1
2
10 45
1024 4.4%
×=≈
10
3
1
2
10 120
1024 11.7%
×=≈
10
4
1
2
10 210
1024 20.5%
×=≈
10
5
1
2
10 252
1024 24.6%
×=≈
10
6
1
2
10 210
1024 20.5%
×=≈
10
7
1
2
10 120
1024 11.7%
×=≈
10
8
1
2
10 45
1024 4.4%
×=≈
10
9
1
2
10 10
1024 1.0%
×=≈
10
10
1
2
10 1
1024 0.1%
×=≈
44 binomial theorem
Number of Heads Probability
0
1
2
3
4
5
6
7
8
9
10
If the coin is biased—say the chances of tossing a head
are now only 1/3—then a different binomial distribu-
tion would be obtained. For example, the probability
of attaining precisely eight heads among 10 tosses is
now only , about 0.3 percent.
In general, if pdenotes the probability of success,
and q= 1– pis the probability of failure, then the
binomial distribution is given by the formula
, the probability of attaining precisely k
successes in nruns of the experiment. This quantity is
the kth term of the binomial expansion formula from
the
BINOMIAL THEOREM
:
This explains the name of the distribution.
The binomial distribution has mean value (
EXPECTED
VALUE
) , which equals np. (To see this,
differentiate the formula with
respect to p.) The standard deviation is σ= .(See
STATISTICS
:
DESCRIPTIVE
.)
The Poisson Distribution
It is difficult to calculate the binomial distribution if nis
very large. Mathematicians have shown that the bino-
mial distribution can be well approximated by the
NOR
-
MAL DISTRIBUTION
for large values of n, provided the
value pis neither extremely small nor close to one. For
these troublesome values of p, S
IMÉON
-D
ENIS
P
OISSON
showed in 1837 that the values —with µ= np, for
k= 0, 1, 2, …—provide a sequence of values close to
the values one would expect from the binomial distribu-
tion. The distribution provided by these approximate
values is called a Poisson distribution.
The Geometric Distribution
Alternatively one can ask: what is the probability that
the first success in a series of experiments occurs on the
nth trial? If pis the probability of success and q= 1 – p
is the probability of failure, then one obtains a first
success on the nth experiment by first obtaining n– 1
failures and then a success. The probability of this
occurring is: P(n) = pqn–1. The distribution given by
this sequence of probability values (for n= 1, 2, 3, …)
is called the geometric distribution. It has mean µ= 1/p
and standard deviation σ= √
–
q/p. The geometric distri-
bution is a special example of Pascal’s distribution,
which seeks the probability Pk(n) of the kth success
occurring on the nth trial.
See also
HISTORY OF PROBABILITY AND STATISTICS
(essay).
binomial theorem (binomial expansion) The identi-
ties (x+ a)2= x2+ 2xa + a2and (x+ a)3= x3+ 3x2a+
µk
k! e–µ
√npq
pq n
kpq
nknk
k
n
+
()
=
−
=
∑
0
µ=
−
=
∑kn
kpq
knk
k
n
0
pq npq npq npq n
npq
nnn n n
+
()
=
+
+
++
−−
01 2
01122 0
L
n
kpq
knk
−
10
8
1
3
2
30 003
82
≈.