Statistics

Probability underlies many of the important methods of inferential statistics: (random sample of data taken from a population to describe  about the population using: testing of statistical hypothesis or parameter estimation approaches). Or, Relative frequency of the possible occurrence in an event.

Event

In considering probability, we deal with procedures (such as rolling a die, answering a multiple-choice test question, or undergoing a test for drug use) that produce outcomes.

Example

An example event is defined as:

Two balls in same colour and one ball in another colour.

Simple Event 

An outcome or event that cannot be further broken down into simpler components.

Sample Space

For a particular procedure, consists of all possible simple events

Notations

1. P denotes a probability
2. A, B, and C denote specific events
3. P(A) denotes the probability of event A occurring
4. The probability values are expressed as numbers between 0 and 1 inclusive.

Venn Diagrams

Show all possible logical relations between a finite collection of sets – events (aggregation of things)

If an event occurs and we call it A, the complement of this event A, is denoted Ā that Ā consists of all outcomes in which A does not occur. A and Ā are mutually exclusive.

Rules:

P(A)+P( Ā )=1

P( Ā )=1-P(A)

P(A)=1-P( Ā )

Approaches to probability

Classical approach
Relative frequency approximation
Subjective approach 

Classical approach to probability
Rule:

For a given procedure resulting in n simple event, if we know that each of simple event has an equal chance to occur , the probability P(A) is:

Example:

Randomly selecting an integer from [1,10] and what is the probability of event P(A) ?

Relative Frequency Approximation

Rule:

Conduct a procedure with limited trials repeated, counting the number of times that an event actually occurs during the trials

For an Event A, P(A) is then estimated as followings

Example

In 2011 there were 203,950 separate casualties  on road in UK, whilst for the same year there were 28,500,000 cars registered. Use the relative frequency approach find the probability?

Subjective approach to probability 

Rule:

For a specific event, e.g., A, the probability P(A) is estimated by using knowledge of the relevant circumstance.

For example, the probability of rain is estimated based on expert knowledge and meteorology information .

Example

The probability that a student watches film A is 0.4 and for film B 0.5 and for both films is 0.2. Find the probability of a random watches film A or film B.

P(A)=0.4

P(B)=0.5

P(A∩B)=0.2

P(AUB)=P(A)+P(B)-P(A∩B)=0.4+0.5-0.2=0.7

Conditional probability 

Example

Suppose 70% of students at a college pass core 6, 55% pass core 8 and 45% pass for both. If a randomly selected student passed core 6, what is the probability of student also passed core 8?

Law of Total Probability

Bayes Theorem