Download Review of Elementary Probability Definitions and Properties

FOUNDATION MATHEMATICS 042
Review of Elementary Probability Definitions and Properties
(Random) experiment. A method by which observations are made, e.g. one can have the
experiment of rolling a die once, to observe the number of dots on its uppermost face.
Outcome. A possible observation of an experiment, e.g. 1 is an outcome of the above experiment.
Sample space. The set of all outcomes of an experiment, e.g. {1, 2, 3, 4, 5, 6} is the sample
space of our example experiment.
Event. A set of outcomes of a random experiment, e.g. the event that one rolls an even
number is the set {2, 4, 6}. A simple event is a set containing a single outcome of a random
experiment.
With the following properties of sets and probability functions it is recommended
that you draw for yourself corresponding Venn diagrams.
Probability function. P is a probability function on a sample space S if
• 0 ≤ P (A) ≤ 1 for every event A of S;
• P (S) = 1; and
• P (A ∪ B) = P (A) + P (B) for every pair of disjoint events A, B of S.
A and B are disjoint if their intersection is empty, i.e. A ∩ B = ∅. Note that it follows from
above that
• P (A ∩ B) = 0 if A ∩ B = ∅.
Suppose Xi represent the outcomes of a random experiment. Then, to show that P is a
probability function on its sample space S, it is enough to show that
(i) P (Xi ) ≥ 0 for each outcome Xi ∈ S, and
P
(ii)
i P (Xi ) = 1.
Complementary event. The complement of an event A in a sample space S, is the set of
all outcomes in S that are not in A. We denote this complementary event of A, by A (or
sometimes by A0 ).
De Morgan’s Laws.
• A∪B =A∩B
• A∩B =A∪B
1
Correspondence ∪ ↔ or, ∩ ↔ and.
• A ∪ B is the event that A or B occurs.
• A ∩ B is the event that A and B occur.
Mutually exclusive. Two events A, B are mutually exclusive if A, B are disjoint as sets.
Independent. Two events A, B are independent if the probability of one event occurring is
unaffected by whether or not the other event has occurred.
Conditional probability. It may happen that the sample space S is effectively reduced to
a subset B of S. In this case we may say the probability of an event A given B and write
P (A | B). B here is the “condition”.
Further properties of probability functions. Let P be a probability function on the
sample space S. Then we have the following properties where A, B are events in S.
• Complementary events A and A satisfy:
P (A) = 1 − P (A)
• The conditional probability P (A | B) is given by:
P (A | B) =
P (A ∩ B)
P (B)
• If A and B are independent events then
P (A | B) = P (A) and P (B | A) = P (B)
so that we have
P (A ∩ B) = P (A | B).P (B),
= P (A).P (B)
by rearranging the conditional probability rule
• For any events A, B in S,
P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
Hence, if A, B are mutually exclusive, so that P (A ∩ B) = 0, then
P (A ∪ B) = P (A) + P (B)
2