Combinatorial Probabilities Cheat Sheet Page 2
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2Math 408, Actuarial Statistics I
A.J. Hildebrand
Binomial coefficients
!
Definition: For
= 1 2
and
= 0 1
,
=
.
!(
)!
(Note that, by definition, 0! = 1.)
Alternate notations:
or
(
)
(
1)
(
+ 1)
Alternate definition:
=
.
!
(This version is convenient for hand-calculating binomial coefficients.)
Symmetry property:
=
Special cases:
=
= 1
=
=
0
1
1
Binomial Theorem: ( + ) =
=0
Binomial Theorem, special case:
(1
)
= 1
=0
Combinatorial Interpretations:
represents
1. the number of ways to select
objects out of
given objects (in the sense of
unordered samples without replacement);
2. the number of -element subsets of an -element set;
3. the number of -letter HT sequences with exactly
H’s and
T’s.
Binomial distribution: Given a positive integer
and a number with 0
1, the
binomial distribution (
) is the distribution with density (p.m.f.) ( ) =
(1
)
, for
= 0 1
.
2
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