Probability is a foundational subject
‘Mathematics is the logic of certainty, probability is the
logic of uncertainty’ – Blitzstein and Hwang.
Probability
Applications:
Statistics, physics, chemistry, biology, computer science, finance,
data science, political sciences, epidemiology, modeling, Monte
Carlo, operations research, forecasting, prediction. . . .
Probability is unintuitive Good news
We can improve our intuition.
Despite its ubiquity in every aspect of our lives, humans generally
have a bad intuition for the subject. We don’t need intuition. Just apply the logical framework we will
learn in this course.
,Consultant question Definitions and notation
,.÷µµ ¥µ
A sample space, denoted by Ω, is the set of all possible outcomes
You are a medical consultant. A study of the residents of a region of a random experiment.
showed that 20% are smokers. An event, denoted by A, or B, etc., is a combination of outcomes,
The probability of death due to lung cancer, given that a person and a subset of the sample space Ω.
smokes, is ten times the probability of death due to lung cancer, Probability is a function, on the outcomes and events of the
given the person does not smoke. sample space Ω.
If the probability of death due to lung cancer in the region is 0.006, ! This function assigns a numerical value to each outcome and
what is the probability of death due to lung cancer given that a event in the sample space, according to specified rules.
person is a smoker? ! The probability of an event A is denoted P(A).
Note: a sample space may be discrete or continuous.
Examples of sample spaces Examples of events
! Annual rainfall for Adelaide could take any non-negative value:
! Rainfall less than 600mm in a year:
! The number of cars passing a given point on the road in one ! Three cars passing a given point:
hour could take any non-negative integer: ! Obtaining exactly 2 heads:
! The outcome of tossing 3 distinct coins:
,
‘Mathematics is the logic of certainty, probability is the
logic of uncertainty’ – Blitzstein and Hwang.
Probability
Applications:
Statistics, physics, chemistry, biology, computer science, finance,
data science, political sciences, epidemiology, modeling, Monte
Carlo, operations research, forecasting, prediction. . . .
Probability is unintuitive Good news
We can improve our intuition.
Despite its ubiquity in every aspect of our lives, humans generally
have a bad intuition for the subject. We don’t need intuition. Just apply the logical framework we will
learn in this course.
,Consultant question Definitions and notation
,.÷µµ ¥µ
A sample space, denoted by Ω, is the set of all possible outcomes
You are a medical consultant. A study of the residents of a region of a random experiment.
showed that 20% are smokers. An event, denoted by A, or B, etc., is a combination of outcomes,
The probability of death due to lung cancer, given that a person and a subset of the sample space Ω.
smokes, is ten times the probability of death due to lung cancer, Probability is a function, on the outcomes and events of the
given the person does not smoke. sample space Ω.
If the probability of death due to lung cancer in the region is 0.006, ! This function assigns a numerical value to each outcome and
what is the probability of death due to lung cancer given that a event in the sample space, according to specified rules.
person is a smoker? ! The probability of an event A is denoted P(A).
Note: a sample space may be discrete or continuous.
Examples of sample spaces Examples of events
! Annual rainfall for Adelaide could take any non-negative value:
! Rainfall less than 600mm in a year:
! The number of cars passing a given point on the road in one ! Three cars passing a given point:
hour could take any non-negative integer: ! Obtaining exactly 2 heads:
! The outcome of tossing 3 distinct coins:
,
