Probability Theory: Interpretations, Axioms, and Applications, Slides of Engineering Mathematics

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Interpretations of probability

Kolmogorov’s axioms

Midterm

• 22 nd^ Mar, F2~F3, MMW LT

• Bring calculator and blank papers.

• Close-book and close-note exam.

• No make-up exam

• Coverage:

– Lecture 1 to 15.

– Tutorial 1 to 7.

– Homework 1 to 3.

Last week: Conditional probability

• If we are given an addition information that the outcome is in event B, then we can update the likelihood to |AB|/ |B|.
• If Pr(B) is nonzero, then we define the conditional probability of A given B by Pr(A|B) = Pr(AB)/Pr(B).

 A B

At the beginning, the probability of event A is |A| / ||, assuming that all outcomes in  are equally likely.

Disjoint events and independent

events

• If two events E and F are disjoint, i.e., EF=,

– then they cannot occur at the same time.

– Pr(E  F) = Pr(E) + Pr(F)

– Pr(E  F) = 0

• If two event A and B are statistically

independent, then

– Pr(AB) = Pr(A)Pr(B),

– AB is in general not empty.





a E

F

a A

B

The Bayes’ rule

• Let A and B be two events. Suppose that the

probability of B is nonzero.

• Probability of A given B can be computed from

the probability of B given A by

Example: The paradox of HIV test

• There is a test for AIDS.

– If a person is infected, the test will be positive 99%

of the time.

– If a person is not infect, the test will be negative

98% of the time.

• Suppose person A take the test, and the result

is positive. Person A comes from a population

with infection rate 0.5%. What is the chance

that A is infected? This is called a priori probability

INTERPRETATIONS OF PROBABILITY

Classical interpretation

• Usually credited to Laplace.

• All outcomes in the sample space  have the

same probability.

• Pr(E) = |E| / |  |

• It requires that the sample space  is a finite set.

• Calculation of probability reduces to counting.

• The classical interpretation cannot model biased

coin, loaded dice, etc.

Subjective interpretation

• Probabilities are defined by personal

preference, or personal belief.

• Example: “2013年 3 月 1 日 14:47 長遠房屋策略督導委員會

• Used in social science, decision making, etc.

Example of subjective probability

• There are two suspects X and Y in a murder case.

Both of them are on the run. Initially, X and Y

have the same evidence.

• After more detailed investigation, it is known that

the murderer has blood type B.

• We also know that suspect X has blood type B,

but we do not know the blood type of suspect Y.

• It is also known that 10% of the population has

blood type B.

• What is the probability that Y is the murderer

given the new information?

Geometric probability

• A.k.a. “stochastic geometry”

• The sample space is a geometric object, like a

line segment, circle, square, sphere, etc.

• Pick a point randomly in the sample space.

The probability that the chosen point falls

within a certain region is directly proportional

to the area/volume of the region.

– This is the uniform distribution on the geometric

object.

Example

• Pick a point at random on a circle of radius 1.

• For 0<r<1, this point falls within a circle of

radius r with probability  r^2 / .

• The probability of picking

particular point is zero,

because a point as zero area.

Circle of radius 1

Probability measure

• Regardless of the interpretations, the calculations of

probabilities must satisfy some basic rules.

• Three of the very basic rules are identified by Russian

mathematician Kolmogorov as the axioms of probability.

• In Kolmogorov’s formulation, we assign a real number to an

event. A probability measure is a function which accepts an event as an input and output a real number.

• As we saw in the example of geometric probability, it is sometime more convenient to assign probabilities to events instead of points in the sample space.
• We construct a probability model after assigning probability

to sufficiently large number of basic events. The other probability events can be computed by union, intersection, and complements of the basic events.

Kolmogorov’s axioms

• Let  be a sample space, which may be finite or

infinite.

• A probability measure assigns real numbers to

some events in,  satisfying the following

axioms:

1. Pr(E) is a real number between 0 and 1.

2. Pr() = 1

3. For any sequence of pairwise disjoint event E 1 , E 2 , E 3 , …, we have

Pr (E 1  E 2  E 3  …) = Pr(E 1 ) + Pr(E 2 ) + Pr(E 3 ) + …