Probability and Stochastic Processes

Chapter 1 Experiments, Models, and Probabilities

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We will learn
• the logic of probability theory
• how to apply probability theory to solving engineering
problems

The abstract mathematics of probability consists definitions,

axioms, and theorems:
• Definitions establish the logic of probability theory
• Axioms (only three) are facts that we have to accept without
proof
• Theorems are consequences that follow logically from
definitions and axioms
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 1.1 Set Theory
• Venn Diagrams
• Universal Set/Empty Set

• Union/Intersection

• Complement
• Mutually Exclusive/Collectively Exhaustive

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 Set Theory
The mathematical basis of probability is the theory of sets.
• A ⊂ B iff x ∈ A ⇒ x ∈ B

• A = B iff A ⊂ B and B ⊂ A

• universal set S
• null set φ

• x ∈ A ∪ B iff x ∈ A or x ∈ B
• x ∈ A ∩ B iff x ∈ A and x ∈ B

• x ∈ Ac iff x ∈ A
• x ∈ A − B (the difference between A and B) iff x ∈ A and x ∈ B
(A − B = A ∩ B c and Ac = S − A)

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 Set Theory
• A collection of sets A1 , A2 , · · · , An is mutually exclusive iff
Ai ∩ Aj = φ, i = j.
• A and B are disjoint iff A ∩ B = φ.

• A collection of sets A1 , A2 , · · · , An is collectively exhaustive iff

∪ni=1 Ai = S.
• De Morgan’s Law: (A ∪ B)c = Ac ∩ B c .

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 1.2 Applying Set Theory to Probability
The mathematics of probability can be abstract, just as we defined sets and
set operations without any reference to physical phenomena.

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 What is Probability
• a number between 0 and 1.
• relative frequency of occurrence?

• subjective belief?

• The modern axiomatic approach to probability theory assumes a set of

axioms about probability.

• In many cases, we assign probabilities based on intuitive or physical

reasoning. (= modeling)

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 Experiments
• Procedure + Observations
• Real Experiments are TOO complicated

• Instead we analyze/develop models of experiments

– A coin flip is equally likely to be H or T

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 Example 1.1
An experiment consists of the following procedure, observation and model:
• Procedure: Flip a coin and let it land on a table.

• Observation: Observe which side (head or tail) faces you after the coin
lands.
• Model: Heads and tails are equally likely. The result of each flip is
unrelated to the results of previous flips.

Examples 1.2 and 1.3

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 Definitions 1.1 Outcome, 1.2 Sample Space, 1.3 Event
An outcome of an experiment is any possible observation of that experiment.
The sample space of an experiment is the (finest-grain, mutually exclusive,
collectively exhaustive) set of all possible outcomes.
An event is a set of outcomes of an experiment.

Examples 1.4, 1.5

Examples 1.6–1.8

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 Correspondences
Set Algebra Probability
set event
universal set sample space
element outcome

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 Example 1.9
Flip four coins, a penny, a nickel, a dime, and a quarter. Examine the coins in
order (penny, then nickel, then dime, then quarter) and observe whether each
coin shows a head (h) or a tail (t). What is the sample space? How many
elements are in the sample space?
......................................................................
The sample space consists of 16 four-letter words:

{tttt, ttth, ttht, . . . , hhhh}

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 Definition 1.4 Event Spaces
• An event space is a collectively exhaustive, mutually exclusive set of
events.

• The sets B1 , B2 , · · · , Bn forms a partition of the sample space S

if (i) ∪ni=1 Bi = S and (ii) Bi ∩ Bj = φ, i = j.

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 Event Spaces
• Example 1.10: For i = 0, 1, 2, 3, 4,

Bi = {outcomes with i heads}

• Each Bi is an event containing one or more outcomes:

• The set B = {B0 , B1 , B2 , B3 , B4 } is an event space.

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 Theorem 1.2
• For an event space B = {B1 , B2 , . . .} and any event A, let Ci = A ∩ Bi .

• For i = j, the events Ci and Cj are mutually exclusive (Ci ∩ Cj = φ),

and

•
A = C1 ∪ C2 ∪ · · ·

Example 1.11

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 Axioms of Probability
A probability measure P [·] is a function that maps events in the sample space
to real numbers that satisfies the following axioms:
Axiom 1 P [A] ≥ 0 for every event A.

Axiom 2 P [S] = 1.
Axiom 3 For any countable collection A1 , A2 , . . . of mutually exclusive
events
∞
P [∪∞
i=1 Ai ] = P [Ai ]
i=1

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 Consequences of the Axioms
• Theorem 1.4: If A = A1 ∪ A2 ∪ · · · ∪ Am and Ai ∩ Aj = φ for i = j,
then
 m
P [A] = P [Ai ]
i=1

• Proof: Consider a sequence of events A1 , A2 , . . ., where A1 = S,

Ai = φ for i > 1. Since S = ∪∞i=1 Ai , we have from Axiom 3 that
∞
 ∞

P [S] = P [Ai ] = P [S] + P [φ]
i=1 i=2

implying that P [φ] = 0. Theorem 1.4 follows from Axiom 3 by defining

Ai = φ for i > m.

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 Consequences of the Axioms
• Theorem 1.3: Given events A and B such that A ∩ B = φ, then

P [A ∪ B] = P [A] + P [B]

• Theorem 1.5: Given B = {s1 , s2 , · · · , sn },

m

P [B] = P [{si }]
i=1

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 Equally Likely Outcomes
• Theorem 1.6: For an experiment with sample space
S = {s1 , s2 , · · · , sn } in which each outcome si is equally likely,
1
P [si ] = , 1≤i≤n
n

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 Consequences of the Axioms
Theorem 1.7: The probability measure P [·] satisfies
• P [φ] = 0.

• P [Ac ] = 1 − P [A].

• For any A and B,

P [A ∪ B] = P [A] + P [B] − P [A ∩ B]

• If A ⊂ B, then P [A] ≤ P [B].

Theorem 1.8: For any event A and event space {B1 , B2 , . . . , Bm },

m

P [A] = P [A ∩ Bi ].
i=1

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 Defintion 1.6 Conditional Probability
• The conditional probability of the event A given the occurrence of the
event B is
P [AB]
P [A|B] =
P [B]

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 Theorem 1.9
A conditional probability measure P [A|B] has the following properties that
correspond to the axioms of probability.
Axiom 1 P [A|B] ≥ 0.

Axiom 2 P [B|B] = 1.
Axiom 3 If A = A1 ∪ A2 ∪ · · · with Ai ∩ Aj = φ for i = j, then

P [A|B] = P [A1 |B] + P [A2 |B] + · · ·

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 Theorem 1.10 Law of Total Probability
• If B1 , B2 , . . . , Bm is an event space and P [Bi ] > 0 for i = 1, . . . , m,
then
m
P [A] = P [A|Bi ]P [Bi ]
i=1

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 Theorem 1.11 Bayes’ Theorem
•
P [A|B]P [B]
P [B|A] =
P [A]
• For an event space B1 , B2 , . . . , Bm ,
P [A|Bi ]P [Bi ]
P [Bi |A] = m
i=1 P [A|Bi ]P [Bi ]

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 Definition 1.7 Two Independent Events
Events A and B are independent if and only if

P [AB] = P [A]P [B]

......................................................................
From Definition 1.7, it follows that

P [A|B] = P [A] and P [B|A] = P [B] if P [A] = 0 and P [B] = 0.

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 Definition 1.8 Three Independent Events
A1 , A2 , and A3 are independent if and only if
• A1 and A2 are independent.

• A2 and A3 are independent.

• A1 and A3 are independent.

• P [A1 ∩ A2 ∩ A3 ] = P [A1 ]P [A2 ]P [A3 ].

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 Fundamental Principle of Counting
• Experiment A has n possible outcomes,
• Experiment B has k possible outcomes,

• There are nk possible outcomes when you perform both experiments.

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 Permutations
• k-permutation: an ordered sequence of k distinguishable objects
• (n)k = no. of k-permutations of n dist. objects.
n!
(n)k = n(n − 1)(n − 2) · · · (n − k + 1) =
(n − k)!

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 Combinations
• Pick a subset of k out of n objects.
• Order of selection doesn’t matter

• Each subset is a k-combination

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 How Many Combinations
n
• k = “n choose k”
• Two steps for a k-permutation:
1. Choose a k-combination out of n objects.
2. Choose a k-permutation of the k objects in the k-combination.
   
n n n!
(n)k = k! =
k k k!(n − k)!

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 Problem 1.8.6
A basketball team has
• 3 pure centers, 4 pure forwards, 4 pure guards

• one swingman who can play either guard or forward.

A pure player can play only the designated position. How many lineups are
there (1 center, 2 forwards, 2 guards)

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 Problem 1.8.6 Solution
Three possibilities:
1. swingman plays guard: N1 lineups
2. swingman plays forward N2 lineups
3. swingman doesn’t play. N3 lineups

N = N1 + N2 + N3
   
3 4 4
N1 = = 72
1 2 1
   
3 4 4
N2 = = 72
1 1 2
   
3 4 4
N3 = = 108
1 2 2

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 1.9 Independent Trials
• Independent trials: (i) each time, a success occurs with constant
probability p; otherwise, a failure occurs with probability 1 − p, (ii) the
results of each trial is independent of the results of all other trials.
• For n independent trial,
Each outcome with k successes has probability pk (1 − p)n−k
n
There are k outcomes that have k successes
n k
• P [Sk,n ] = k p (1 − p)n−k : Binomial distribution

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 Multiple Outcomes
• n independent trials
• r possible trial outcomes (s1 , . . . , sr )

• P [sk ] = pk

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 Multiple Outcomes (2)
• Outcome is a sequence:
– Example: s3 s4 s3 s1

P [s3 s4 s3 s1 ] = p3 p4 p3 p1 = p1 p23 p4
= pn1 1 pn2 2 pn3 3 pn4 4

• Prob depends on how many times each outcome occurred

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 Multiple Outcomes (3)
• Ni = no. of time si occurs

P [N1 = n1 , . . . , Nr = nr ] = M pn1 1 pn2 2 · · · pnr r

M = Multinomial Coefficient
     
n n − n1 n − (n1 + n2 ) nr
= ···
n1 n2 n3 nr
n!
=
n1 !n2 ! · · · nr !

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 Example 1.47 (MATLAB Demo)
Simulate the testing of 100 microprocessors as described in Example 1.43.
Your output should be a 4 × 1 vector X such that Xi is the number of grade i
microprocessors.

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 Virtual Laboratories
• Buffon’s Coin Experiment
• Buffon’s Needle Experiment

• Binomial Coin Experiment

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 Chapter Summary
• Sample space, event, and outcome
• Probability measure P [A]

• Conditional probability P [A|B]

• Independent events
• Counting methods

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