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Probability & Statistics
Handout
Dr. Matthias Sachs
Last update: April 14, 2024
1 Fundamentals
Many natural situations involve some notion of randomness or unpredictability. You might think of
rolling a dice, or flipping a coin, or checking the Google share price. In this course we will refer to such
random procedures as random experiments or random trials. In this section we introduce the
following basic concepts that are used to describe such random experiments mathematically. These
are the notions of:
1. a sample space or outcome space, the set of all possible outcomes of the random experiment.
Typically, we denote this set as Ω. Each element ω ∈ Ω is called a sample or an outcome. The
experiment always produces exactly one outcome.
2. events, which are subsets of the outcome space, i.e., each event is a subset A, A ⊆ Ω. If the
random experiment produces an outcome ω ∈ A then we say that the event A occurs.
3. a probability distribution P, and, in particular the probability P(A) of an event A ⊆ Ω.
Example 1.1 (Rolling a 6-sided die). If our random experiment is rolling a 6-sided die, then a typical
choice of the outcome space Ω is
• Ω = {1, 2, 3, 4, 5, 6}
For this choice of outcome space, examples of events are
• “roll an odd number” = {1, 3, 5}
• “roll number ≤ 2” = {1, 2}
We commonly denote the probability of an event A ⊆ Ω as P(A), where mathematically P is a
function on subsets of Ω, which satisfies certain properties and as such is referred to as a probability
distribution (see more in 1.2). The following definition introduces a special instance of such a
probability distribution.
1
Definition 1.2 (Uniform Probability distribution on finite outcome spaces). Consider a finite outcome
space Ω and events A ⊆ Ω. If outcomes are equally likely, then
P(A) =
|A|
|Ω|
,
and P is referred to as the uniform probability distribution on Ω. Here, B ⊆ Ω, |B| denotes the
magnitude of the set B, i.e., |B| = “number of elements in B”.
Example 1.1 (Continued). If the die is fair, then each of the numbers 1, . . . , 6 is equally likely to be
rolled and thus the probability distribution is the uniform probability distribution on Ω = {1, . . . , 6}.
In particular,
P(“Roll even number”) =
|{2, 4, 6}
|{1, 2, 3, 4, 5, 6}|
=
1
2
.
Example 1.3 (Rolling two fair dice). As the outcome space we can take the sample space
Ω =
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
,
where the first number in each ordered pair corresponds to the number rolled by the “first” die and the
the second number in each ordered pair corresponds to the number rolled by the “second” die. Since
both dice are fair, the probability distribution on Ω is uniform. Thus, we can compute probabilities
of events as follows:
1.
P(“Both dice show 1”) =
|(1, 1)|
|Ω|
=
1
36
.
2.
P(“Sum of the rolled numbers is 5”) =
|(1, 4), (2, 3), (3, 2), (4, 1)|
|Ω|
=
4
36
=
1
9
.
3.
P(“Number rolled with first die < number rolled with second die”) =
(36 −6)/2
|Ω|
=
15
36
.
1.1 Interpretation of probability
If the outcome space is a countable set, then
• P(A) = 0 means that the event A never occurs.
• P(A) = 1 means that the event A always occurs.
2
Example 1.3 (Continued). When rolling two fair die, we have
1.
P(“Sum of the rolled numbers > 12”) =
|∅|
|Ω|
= 0.
2.
P(“Rolled number on first die = 12.5”) =
|∅|
|Ω|
= 0.
3.
P(“Sum of the rolled numbers ≤ 12.5”) =
|Ω|
|Ω|
= 1.
Question: what is the interpretation of an event probability P (A) = p with 0 < p < 1?
Definition 1.4 (Frequentist interpretation of probability). Consider a random experiment with out-
come space Ω, and let A ⊆ Ω. Let
P
n
(A) = “proportion of times that A occurs in n trials”,
then
P(A) ≈ P
n
(A) for sufficiently large n.
Example 1.5 (Fair coin toss). Consider tossing a fair coin, i.e., Ω = {H, T }, and P({H}) = P({T }) =
1
2
. When repeating the experiment, we observe the following sequence of outcomes:
Trial number n 1 2 3 4 5 6 7 8 9 10 . . .
Outcome ω T T H H T H T H H T . . .
P
n
({T }) 1
2
2
2
3
2
4
3
5
3
6
4
7
4
8
4
9
5
10
. . .
1.2 Axiomatic definition of probability
A probability distribution “P” can be understood as function that maps subsets of the outcome space
Ω to values in the interval [0, 1], e.g.,
P : {A : A ⊆ Ω} → [0, 1].
Example 1.6.
Ω = {1, 2, 3},
then
P :
∅, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}
→ [0, 1].
Definition 1.7. Let B ⊆ Ω . The collection B
1
, . . . , B
n
⊆ Ω is called a partition of B, if
3
Figure 1: Left figure: plot of the P
n
({T }) for the sequence of trials shown in the above table. Right
plot: if we repeat the sequence of random trials, we may obtain a different sequence of trial outcomes,
but the proportion P
n
({T }) will ultimately again tend to 1/2 as n → ∞.
1. B = B
1
∪ B
2
∪ . . . B
n
.
2. The sets B
1
, . . . , B
n
are pairwise disjoint, i.e., B
i
∩ B
j
= ∅ if i 6= j.
Definition 1.8. Let Ω be a non-empty finite set. A function P : {A : A ⊆ Ω} → R is called a
probability distribution if it satisfies the following Rules of Probability:
(i) P(A) ∈ [0, 1] for every event A ⊆ Ω;
(ii) P(Ω) = 1;
(iii) Addition Rule: if A
1
. . . , A
n
is a partition of A ⊆ Ω, then
P (A) =
n
X
k=1
P (A
k
).
If Ω is a finite (or more general a countable set), probability distributions are usually defined by
specifying the probability for each individual outcome:
Example 1.9. On the outcome space Ω = {1, 2, 3} a possible choice for a probability distribution
would be
ω 1 2 3
P({ω})
1
2
1
3
1
6
Example 1.10 (Bernoulli distribution with success probability p). .
Let Ω = {0, 1}, and
4
ω 0 1
P({ω}) 1 − p p
where 0 ≤ p ≤ 1.
Note: If p =
1
2
, then this is an abstract description of a fair coin flip.
Example 1.11. Consider a bag containing
• 1 yellow ball
• 2 green balls
• 3 red balls
• 4 blue balls
We pick a ball at random from the bag and note the color of the ball, i.e.,
Ω = {yellow, green, red, blue}.
The probabilities of picking a ball with color ω, is proportional to the balls of color ω, thus
ω yellow green red blue
P({ω})
1
10
2
10
3
10
4
10
Instead of providing values of individual outcomes in a tabularized form, we can just specify these by
a function p : Ω → [0, 1], termed probability mass function.
Definition 1.12 (Probability Mass function). A function P : Ω → [0, 1] satisfying
X
ω∈Ω
P (ω) = 1,
is referred to as a probability mass function. If for a given probability distribution P on Ω, we have
P (ω) = P({ω}) for all ω ∈ Ω,
then P is called the probability mass function of P.
The following lemma 1.13 shows that every probability mass function defines a probability distribution.
In particular, the lemma also retrospectively justifies specifying probability distributions in tabularised
form as we did in examples 1.9 - 1.11 .
Lemma 1.13. Let p : Ω → [0, 1] with
P
ω∈Ω
p(ω) = 1. Then the function P defined by
P(A) :=
X
ω∈A
p(ω) for all events A ⊆ Ω,
is a probability distribution on Ω. (In particular, P({ω}) = p(ω) for all ω ∈ Ω.)
5
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