2.2 Random Events and Probability
Ulrich Hoensch
Monday, January 11, 2010
Basic Definitions
I
A random experiment satisfies the following conditions.
1. A single element is selected from a set S. The selection is
called a trial; the set is called the sample space; the elements
of S are called outcomes.
2. For any trial, it is not known which particular outcome will
occur.
3. The experiment can be repeated under similar conditions.
I
An event A is a subset of the sample space: A ⊂ S.
1. The empty set ∅ is the impossible event.
2. The event A ∪ B consists of all outcomes that are in A or B
(or both).
3. The event A ∩ B consists of all outcomes that are in A and B.
4. The event Ac is called the complement of A and consists of
all outcomes that are not in A.
5. The events A and B are called mutually exclusive if
A ∩ B = ∅ (they have no outcomes in common).
Example
I
Random experiment: roll a six sided die;
I
S = sample space = numbers on die = {1, 2, 3, 4, 5, 6};
I
event of getting an even number = A = {2, 4, 6};
I
event of getting an odd number = B = {1, 3, 5};
I
event of getting a prime number = C = {2, 3, 5}.
Then,
I
A∩B =
I
A∪B =
I
A∩C =
I
B ∪C =
I
Ac =
I
Cc =
Classical Definition of Probability
Definition 2.2.3
If the sample space S consists of finitely many outcomes which are
all equally likely to occur, then the probability of the event A is
P(A) =
|A|
number of outcomes in A
=
.
number of outcomes in S
|S|
(|A| denotes the number of elements in a set A; |A| = ∞ if A is
infinite.)
Example
If we roll a fair (balanced) die, then for the events defined in the
previous example,
I
P(A) =
I
P(C ) =
I
P(A ∩ C ) =
Frequency Definition of Probability
Definition 2.2.4
Let n(A) be the number of times the event A occurs in n trials.
Then we define the probability of A as
n(A)
.
n→∞ n
This definition of probability can be used in computer simulations
of random experiments. The graph below shows a computer plot
of the proportion of heads when flipping a fair coin n times.
P(A) = lim
1.0
0.8
0.6
0.4
0.2
0.0
1
10
100
1000
Axiomatic Definition of Probability
Definition 2.2.5
Let S be a sample space. The probability P is a real-valued
function that assigns to each event A a number P(A) and that
satisfies the following conditions.
1. P(A) ≥ 0;
2. P(S) = 1;
3. If A1 , A2 , . . . is a sequence of pairwise disjoint events (i.e.
Ai ∩ Aj = ∅ for any i 6= j), then
!
∞
∞
[
X
Ai =
P(Ai ).
P
i=1
i=1
The last property is known as countable additivity. We use
Definition 2.2.5 as our “official” definition of probability. It is due
to A.N.Kolmogorov (1933).
Basic Properties of Probability
Suppose A, B are events in the sample space S. Then:
1. P(Ac ) = 1 − P(A). In particular P(∅) = 0.
2. If A ⊂ B, then P(A) ≤ P(B).
3. P(A ∪ B) = P(A) + P(B) − P(A ∩ B). In particular, if A and
B are mutually exclusive, then P(A ∪ B) = P(A) + P(B).
Example
A die is loaded in such a way that if the number i (i = 1, 2, . . . , 6)
shows up, then P({i}) = Ki, where K is a constant.
I Find the value of K .
I
Find the probability that a prime number or an even number
shows up.
Homework Problems for Section 2.2 (Points)
p.60-63: 2.2.3 (4), 2.2.8 (2), 2.2.17 (5), 2.2.19 (3).
Homework problems are due at the beginning of the class on
Friday, January 15.