Lecture 2

Introduction to Probability

Motivating examples from industry

Concepts
    ● Random experiments
    ● Sample space
    ● Sample point
    ● Discrete and non-discrete events
    ● Certain and Impossible Events
    ● Set operations on events

Probability in practice

Key in modelling phenomena

Industry case studies

Oil industry

Smart grid

Monitoring (equipment, people, environment)

Self-driving vehicles

Artificial Intelligence and Machine Learning

Smart Grid

Smart grids can improve energy efficiency, motivateand educate consumers.

As more people adopt smart meters, utilities find new opportunities through the data these devices collect.
    Power companies can use this information to:
    Forecast energy demand (and reduce waste or outage)
     – Affect customer usage patterns (to improve efficiency andcost)
     – Prevent power outages (by reacting better to suddenchanges in the grid)
     – Reduce the need to build new power plants (help savemoney and environment)

Data Analytics for Smart Grid

Distribution monitoring
    – According to IBM Big Data & Analytics Hub, a modernized grid allowed one US city to instantly restore power to half of its affected residents after a severe windstorm.
      They went from 80,000 without power to less than 40,000 within two seconds.

Resource allocation
    – Weather forecasting => choose which energy sources will be most efficient.
    – Wind and solar power are heavily weather dependent, so this type of energy should be spared if the forecast suggest less generation (less wind or less sun)
    – Use smart grid data to predict peak load times and reduce demand.

Random Experiments

Experiments in science and engineering
    − Measuring voltage over resistor
    − Measuring speed of light

Controlling all conditions of experiments ensures essentially identical experimental outcomes

Many natural phenomena belong to this class of deterministic events
    − Small differences do occur due to measurement errors or small variability of conditions
    − Example: current linearly increases with increasing voltage over resistor but small deviations are present

In some experiments, we are not able to control the value of all conditions (experimental variables)

Implication: experimental results vary from one performance of the experiment to the next even though the conditions are nearly identical

Such experiments are described as random

Examples of Random Experiments

Coin toss results in head (H) or tail (T)
    − There are two possible outcomes

Formally, coin toss will result in one element from the set {H, T}

Toss a coin twice. What are the possible outcomes?

Coin toss

Four outcomes possible

The set of outcomes is      {HH, HT, TH, TT}

Common mistake is to say that HT and TH are the same outcome!

Examples of Random Experiments

Throwing a die: the result of the experiment is one of the numbers in the set {1, 2, 3, 4, 5, 6}
    − More than one set possible
    − Is number divisible by two? {even, odd}
    − Is number smaller than 3? {true, false}

Manufacturing a product: product is member of the set {faulty, working}

Time between repairs: turbine is repaired every h hours, where, for example, 0<h<1000

Sample Space and Sample Point

Set consisting of all possible outcomes of a random experiment is called a sample space.
    − This set is usually denoted by letter S

Each outcome is called a sample point.

If a sample set has finite number of points, it is called a finite sample space
    − Examples: tossing a die (or coin); number of cars on the parking lot; number of faulty devices manufactured in a month;

Sample Spaces

If a sample set has as many points as there are natural numbers, it is called a countably infinite sample space
    − Examples: number of coin tosses until tail is obtained; number of die tosses until six is obtained

If a sample space has as many points as there are in interval on the real axis, it is called a non-countably infinite (continuous) sample space.
    − Examples: analogue voltage level can take any value from some a range; speed can have a range of values from zero to maximum speed

Discrete vs. nondiscrete space

Discrete sample space: if sample space is finite or countably infinite it is said to be discrete

Non-discrete sample space: if sample space is non-countably infinite it is said to be nondiscrete or continuous

Digital measurements: we take analogue quantity from nature (such as speed, temperature, voltage, current etc.) and discretise it (convert to finitely many levels) for computer processing

Events

Event is a subset A of the sample space S
    − it is a set of possible outcomes
    − Example: Die toss which produces even number, A={2,4,6} while S={1,2,3,4,5,6}

If the outcome of an experiment is an element of A, we say that the event A has occurred.
    − Example from above continued: Die toss results in a four, so event A (even die toss) occurred.

Certain and impossible events

Sample space S represents the sure or certain event since an element of S must occur by definition
    − Example: Die toss must result in one of six outcomes so S={1,2,3,4,5,6}
    − Example: Sample space for quality of manufactured device {defective, non-defective}

The empty set Ø is called the impossible event because an element of Ø cannot occur
    − Example: Throwing a die cannot result in seven nor any other value except the six numbers from S
    − Example: Device can have only two states

Set operations on two events

A and B are events in sample space S.
Then:
    − A U B is the event “either A or B or both”. A U B is called the union of A and B.
    − A ∩ B is the event “both A and B”. A ∩ B is called the intersection of A and B.
    − A' is the event “not A.” A' is called the complement of A.
    − A - B is the event “A but not B”. A-B is called the difference of A and B or relative complement of B in A. Also, A'=S-A.
    − If the sets corresponding to events A and B are disjoint, or, formally, A ∩ B =Ø, then the events are mutually exclusive (or mutually disjoint).

Examples of union, intersection, complement and difference: discrete sample space

Die is tossed and A represents event that toss results in number larger than three, while B represents even tosses.

Find A, B, their union, intersection, complements, differences and if they are mutually disjoint (mutually exclusive)

Click here for Example solution  

Solution

Then, A={4,5,6} and B={2,4,6}.

Union: A U B={2,4,5,6}

Intersection: A ∩ B={4,6}

Complement: A'={1,2,3} since S={1,2,3,4,5,6}

Difference: A-B={5}; B-A={2}

A and B are not mutually disjoint since A ∩ B is not empty