Download Lecture 2: Basic Probability Rules 2.1 Outcomes, events

Lecture 2: Basic Probability Rules
2.1
Outcomes, events and their probabilities
The main purpose for us to learn some basic probability theory and methodology is to quantify
important concepts such as uncertainty, return and risk related to financial markets.
Two basic terms:
• Outcome — A random experiment, e.g. buying 200 shares of Google stock yesterday, will have
a number of possible outcomes today, none of which can be fully predicted.
• Event — An event is a set of outcomes. Each outcome itself is considered a simple event.
Note that the distinction between an outcome and an event need not be clear-cut, depending on
the convenience in each problem.
Example 2.1 Suppose yesterday’s closing price of Apple stock is $50. For simplicity, assume
that possible values for today’s closing price are $45, $48 and $51. Hence there are three outcomes
denoted by 45, 48 and 51. If you are only interested in the two events “up” and “down”, then up
= {51} and down = {45, 48}. The realistic situation in this example should contain many more
possible outcomes than just three. There can be an infinite number of possible outcomes in some
other cases.
Some relations between/among events:
• A ⊂ B:
If event A occurs, then event B must occur.
• A ∪ B is an event that occurs if A occurs, or B occurs, or both occur.
• A ∩ B is an event that occurs if both A and B occur.
• A ∩ B = ∅:
time.
Two events A and B are mutually exclusive, i.e. they never occur at the same
• Ac is an event that occurs whenever A does not occur.
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• Events A1 , ..., An are said to partition event A if ∪ni=1 Ai = A1 ∪ · · · ∪ An = A and Ai ∩ Aj = ∅
for any i, j = 1, ..., n with i 6= j.
Definition of probability
Probability is a function of various events such as
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(i) 0 ≤ P (A) ≤ 1 for any event A, where P (A) denotes the probability of event A.
(ii) P (Ω) = 1 and P (∅) = 0, where Ω denotes a sure event (i.e. the set of all possible outcomes in
a given problem), and ∅ denotes an impossible event.
(iii) If A1 , ..., An partition A, then P (A) =
Pn
i=1 P (Ai ).
Later, (iii) will be extended to the case of infinite partitions, i.e.
(iii’) If A1 , A2 , ... partition A, then P (A) =
P∞
i=1 P (Ai ).
Example 2.1 (continued) Assume P ($45) = 0.05, P ($48) = 0.8 and P ($51) = 0.15. Then
P (up) = 0.15 and P (down) = 0.85.
2.2
Rules for calculation of probabilities
The following rules play an important role in calculating probabilities.
(R1) (addition rule) For any events A and B, P (A∪B) = P (A)+P (B)−P (A∩B). In particular,
P (A ∪ B) = P (A) + P (B) if A and B are mutually exclusive; and P (A) = 1 − P (Ac ).
(R2) (division rule)
The conditional probability of A given B, denoted by P (A|B), is defined by

P
(A∩B)

if P (B) > 0,
P (B)
P (A|B) =

0
if P (B) = 0.
(R3) (multiplication rule) P (A ∩ B) = P (A) P (B|A). An extension to the case of more than two
events will be discussed later.
(R4) (averaging rule) If A1 , ..., An partition the entire outcome space Ω, then for any event B,
P
P (B) = ni=1 P (Ai ) P (B|Ai ).
Note: Venn diagrams and tree diagrams are very useful tools that help apply addition and multiplication rules respectively. We will illustrate how to use these two kinds of diagrams in class.
Definition of independence
Two events A and B are said to be (statistically) independent if any one of the following
equivalent conditions holds:
(C1) P (A ∩ B) = P (A) P (B);
(C2) P (A|B) = P (A);
(C3) P (B|A) = P (B).
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Note: Independence among more than two events can also be defined. An important fact
is that if A1 , ..., An (n > 2) are assumed to be independent, then P (all of A1 , ..., An occur) =
Q
P (A1 ) · · · P (An ) = ni=1 P (Ai ).
Warning: It is crucial to distinguish between the two different assumptions: “A and B are independent”; “A and B are mutually exclusive”. Explain why they are different via simple examples.
Example 2.2 Let A = {Google’s stock falls today}, and B = {Exxon’s stock falls today}. Assume A and B are independent (reasonable?), P (A) = 0.2, P (B) = 0.6.
(a) The probability that at least one of the two stocks (Google’s and Exxon’s) fall today is given
by 0.2 + 0.6 − 0.2 · 0.6 = 0.68. This probability can also be calculated by 1 − 0.8 · 0.4 = 0.68.
(b) The probability that only one of the two stocks falls today is given by 0.2 · 0.4 + 0.8 · 0.6 = 0.56.
Example 2.3 The allocation of Mary’s portfolio consists of 25% in bonds and 75% in stocks.
Suppose in a one-year period, the probabilities for Mary to make profits in bonds and stocks are
9/10 and 2/5 respectively.
(a) The probability that Mary’s portfolio turns out to be profitable in one year is equal to
3 2
21
4 5 = 40 .
1 9
4 10
+
(b) Given that Mary has a loss in her portfolio in one year, the relative proportion of the loss
in stocks is equal to
probability.
3
4
3
5
21
1− 40
=
18
19 .
Note that we interpret the relative proportion as a conditional
I suggest that you try to work out Example 2.2 using a Venn diagram, and Example 2.3 using
a tree diagram.
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