HOME WORK WEB PAGE FOR CLASS MATH 251
Chapter 4
Introduction to Probability
4.1 Experiments, Counting
Rules and Assigning Probabilities
In discussing
probability, we define an experiment to be a process that generates well
defined outcomes.
|
Experiment Toss a coin Roll a die Play a football game |
Experimental Outcomes Head, tail 1,2,3,4,5,6 Win, lose, tie |
Sample Space
The sample space for an
experiment is the set of all experimental outcomes. Experimental outcomes are
also called sample points.
Counting
Rules: Combinations, and Permutations
Combinations
The number of ways of
picking 
unordered
outcomes from 
possibilities.
Also known as the binomial coefficient or choice number and read " choose ,"
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where 
is a
factorial (Uspensky 1937, p. 18). The
factorial is
defined for a positive integer as
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So, for example, 
.
An older notation for the factorial is 
(Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978;
Conway and Guy 1996).
The special case 
is
defined to have value 
,
consistent with the combinatorial interpretation of there being exactly one way
to arrange zero objects (i.e., there is a single permutation of zero elements,
namely the empty set 
).
For example, there are 
combinations
of two elements out of the set 
,
namely 
, 
, 
, 
, 
,
and 
.
These combinations are known as k-subsets.
Permutations
permutation, also called an "arrangement number" or
"order," is a rearrangement of the elements of an ordered list 
into a
one-to-one correspondence with itself.
The number of permutations on a set of elements
is given by (
factorial; Uspensky 1937, p. 18). For example, there are 
permutations
of ,
namely and
,
and 
permutations
of 
,
namely ,
,
,
,
,
and 
The number
of ways of obtaining an ordered subset of elements
from a set of elements
is given by
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|
(Uspensky
1937, p. 18), where is a factorial.
For example, there are 
2-subsets
of ,
namely , , , , , , 
, 
, , 
, 
,
and 
.
The unordered subsets containing elements
are known as the k-subsets of a given set.
Home
Work
1.
Five different books
are on a shelf. In how many different ways could you arrange them?
2.
a) How many
different arrangements are there of the letters of the word numbers?
b)
How many of those arrangements have b as the first letter?
c) How
many have b as the last letter -- or in any specified position?
d)
How many will have n, u, and m together?
3.
a) How many
different arrangements (permutations) are there of the digits 01234?
b)
How many 5-digit numbers can you make of those digits, in which the first
digit is not 0?
c)
How many 5-digit odd numbers can you make?
4.
How many combinations
are there of 5 things taken 4 at a time?
5.
a) Write all the
combinations of abcd taken 1 at a time.
b) Write their
combinations taken 2 at a time.
c) Write their
combinations taken 3 at a time.
d) Write their
combinations taken 4 at a time.
6. a) A door can be opened only with a
security code that consists of five buttons: 1, 2, 3, 4, 5. A code
consists of pressing any one button, or any two, or any three, or any four, or
all five. How many possible codes are there?
b)
If, to open the door, you must press three codes, then how many possible
ways are there to open the door? Assume that the same code may be repeated.