College Algebra 9th txtbk

Review 565
A sequence is called a geometric sequence, or a geometric progression,
if there exists a nonzero constant r, called the common
ratio, such that
a n
a n1
r
or
for every n 7 1
The following formulas are useful when working with geometric
sequences and their corresponding series:
n1
a n a 1 r
S n a n
1 a 1 r
1 r
S n a 1 ra n
1 r
S a 1
1 r
r 1
r 1
r 6 1
nth-Term Formula
Sum Formula—First Form
Sum Formula—Second Form
Sum of an Infinite Geometric
Series
8-4 Multiplication Principle, Permutations,
and Combinations
A counting technique is a mathematical method of determining
the number of objects in a set without actually enumerating them.
Given a sequence of operations, tree diagrams are often used to
list all the possible combined outcomes. To count the number of
combined outcomes without listing them, we use the multiplication
principle (also called the fundamental counting principle):
1. If operations O 1 and O 2 are performed in order with N 1 possible
outcomes for the first operation and N 2 possible outcomes for the
second operation, then there are
possible outcomes of the first operation followed by the second.
2. In general, if n operations O 1 , O 2 , . . . , O n are performed in order,
with possible number of outcomes N 1 , N 2 , . . . , N n , respectively,
then there are
possible combined outcomes of the operations performed in the
given order.
The symbol n! is read n factorial and 0! is defined to be 1.
A particular arrangement or ordering of n objects without repetition
is called a permutation. The number of permutations of n
objects is given by
P n,n n (n 1) . . . 1 n!
A permutation of a set of n objects taken r at a time is an
arrangement of the r objects in a specific order. The number of permutations
of n objects taken r at a time is given by
P n,r n!
(n r)!
A combination of a set of n objects taken r at a time is an
r-element subset of the n objects. The number of combinations of n
objects taken r at a time is given by
C n,r a n r b P n,r
r!
a n ra n1
N 1 N 2
N 1 N 2 . . . N n
n!
r!(n r)!
0 r n
0 r n
In a permutation, order is important. In a combination, order is not
important.
8-5 Sample Spaces and Probability
The outcomes of an experiment are called simple events if one and
only one of these results will occur in each trial of the experiment.
The set of all simple events is called the sample space. Any subset
of the sample space is called an event. An event is a simple event
if it has only one element in it and a compound event if it has more
than one element in it. We say that an event E occurs if any of the
simple events in E occurs. A sample space S 1 is more fundamental
than a second sample space S 2 if knowledge of which event occurs
in S 1 tells us which event in S 2 occurs, but not conversely.
Given a sample space S {e 1 , e 2 , . . . , e n } with n simple events,
to each simple event e i we assign a real number denoted by P(e i ), that
is called the probability of the event e i and satisfies:
1. 0 P(e i ) 1
2. P(e 1 ) P(e 2 ) . . . P(e n ) 1
Any probability assignment that meets conditions 1 and 2 is said to
be an acceptable probability assignment.
Given an acceptable probability assignment for the simple
events in a sample space S, the probability of an arbitrary event
E is defined as follows:
1. If E is the empty set, then P(E ) 0.
2. If E is a simple event, then P(E ) has already been assigned.
3. If E is a compound event, then P(E ) is the sum of the
probabilities of all the simple events in E.
4. If E is the sample space S, then P(E ) P(S ) 1.
If each of the simple events in a sample space S {e 1 ,
e 2 , . . . , e n } with n simple events is equally likely to occur, then we
assign the probability 1n to each. If E is an arbitrary event in S,
then
P(E )
If we conduct an experiment n times and event E occurs with frequency
f(E ), then the ratio f(E )n is called the relative frequency
of the occurrence of event E in n trials. As n increases, f(E )n usually
approaches a number that is called the empirical probability
P(E ). So f(E )n is used as an approximate empirical probability
for P(E ).
If P(E ) is the theoretical probability of an event E and the experiment
is performed n times, then the expected frequency of the
occurrence of E is n P(E ).
8-6 Binomial Formula
Number of elements in E
Number of elements in S n(E )
n(S )
Pascal’s triangle is a triangular array of coefficients for the expansion
of the binomial (a b) n , where n is a positive integer. Notation
for the combination formula is
a n r b C n,r
For n a positive integer, the binomial formula is
(a b) n a
n
k 0
n!
r!(n r)!
a n k b ank b k
The numbers a n b, 0 k n, are called binomial coefficients.
k