Download Elements of Statistics 1 Sample space and events

AMS 102.7 Spring 2006
Jingyu Zou
Elements of Statistics
Lecture Notes # 12
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Sample space and events
Definition 1.1 Sample Space is the set of all possible outcomes of an experiment, denoted by Ω.
Example 1.2 If the experiment is simply tossing a coin, then Ω = {H, T } where H denotes head
and T denotes tail.
Example 1.3 If we toss a die, then Ω = {1, 2, 3, 4, 5, 6}. Each number denotes a possible point
from a single toss.
Example 1.4 Toss two coins. Ω = {HH, HT, T H, T T }. The outcome will be HH if both coins
come up heads; it will be HT if the first coin comes up head and the second comes up tial; it will
be T H if the first comes up tail and the second comes up head; and it will be T T if both comes up
tails.
Definition 1.5 Any subset E of the sample space Ω is known as an event.
Example 1.6 In example 1.2, if E = {H}, then E is the event that a head appears on the flip of
the coin. Similarly, if E = {T }, then E would be the event that a tail appears.
Example 1.7 In example 1.3, if E = {1}, then E is the event that one appears on the toss of the
die. If E = {2, 4, 6}, then E would be the event that an even number appears on the toss.
Example 1.8 In example 1.4, if E = {HH, HT }, then E is the event that a head appears on the
first coin.
For any two events E and F of a sample space Ω, E ∪ F is the event consisting of all points
which are either in E or in F or in both E and F . That is, the event E ∪ F will occur if either E
or F occurs.
For any two events E and F , E ∩ F is the event consisting of all points which are both in E
and F . That is, the event E ∩ F will occur only if both E and F occurs. E ∩ F can be simply
written as EF . If EF = ∅, then E and F are said to be mutually exclusive.
For any event E, E C is the event consisting of all points in the sample space Ω which are not
in E. That is E C will occur if and only if E does not occur.
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Probability
Consider a sample space Ω. For each event E of the sample space Ω, we assume that a number
P (E) is defined as satisfies the following three conditions:
(1) 0 ≤ P (E) ≤ 1 for any event E
(2) P (Ω) = 1
(3) For any sequence of events E1 , E2 , . . . , En which are mutually exclusive, that is, events for
which Ei ∩ Ej = ∅ when i 6= j, then
P (E1 ∪ E2 ∪ . . . ∪ En ) = P (E1 ) + P (E2 ) + . . . + P (En )
Corollary 2.1 (1) The probability of empty event is 0. That is, P (∅) = 0
(2) If event E is a subset of event F , that is, E ⊂ F , then P (E) ≤ P (F )
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