The Probability Operator

Although the probability operator takes various types of arguments and operands, it always produces a value between 0 and 1. It can be shortened to prob to reduce keystrokes. There are three basic forms of the operator:

  1. Contingency Table Form: applies the general rules of probability
  2. Independent Form: assumes independent events, and
  3. Distribution Form: calculates cumulative, upper-tail or point probabilities of discrete and continuous distributions

Each of the basic forms of the probability operator are described in detail below:

Contingency Table Form: General Rules of Probability

We would like to examine probabilties relating to two categorical variables: Sex and Political Party. There are two sexes: M=Male and F=Female; and three parties: D=Democrat, I=Independent and R=Republican. We first create a contingency table with row and column headings and populated with frequencies:     

        Table←↑'*DIR'('F' 3 2 4)('M' 8 9 12)

* D I  R
F 3 2  4
M 8 9 12

We can now apply the basic rules of probability from the following table:

Rules of Probability (Summary)

Term

Symbol

Condition

Special Formula

General Formula

Primary

Operation

Complement

(not A)

A'

None

P(A') = 1 - P(A)

-

Union

(A or B)

A∪B

Mutually Exclusive

P(A∪B) = P(A) + P(B)

P(A∪B) = P(A) + P(B) - P(A∩B)

+

Intersection

(A and B)

A∩B

Independent

P(A∩B) = P(A)P(B)

P(A∩B) = P(A)P(B|A)

P(A∩B) = P(A) + P(B) - P(A∪B)

×

Conditional

(A if B)

A|B

Independent

P(A|B) = P(A)

P(A|B) = P(A∩B)/P(B)

Mutually Exclusive

P(A|B) = 0

The syntax of the probability operator is:

            [Event1]  relationalFunction prob Table , Event2

Note that the comma  is necessary to separate the array right operand Table from the right argument Event2.

What is the probability that a randomly selected student is not a Republican? 

      ~ prob Table, 'R'
0.57895

What is the probability that a student is both Republican and male?

     'R' ∧ prob Table, 'M'
0.31579
      

What is the probability that a student is either Republican or male?

            'R' ∨ prob Table, 'M'
0.86842
 

What is the probability that a male student is a Republican?

           'R' | prob Table, 'M'
0.41379
 

Independent Form

When two events are independent, only the probabilities of the two individual events are needed, not the entire contingency table.  In that case, the syntax is:

         P(Event1) relationalFunction prob independent P(Event2)

What is the probability of selecting a spade from a deck of 52 cards?

        Spade←13÷52
0.25

What is the probability of selecting an ace?

        Ace←4÷52
0.076923

What is the probability of selecting the ace of spades?

        Ace ∧ prob independent Spade
0.01923

   What is the probability of selecting  an ace or spade?

        Ace ∨ prob independent Spade
0.30769

Distribution Form

The left operand to the distribution form of the probability operator is a discrete or continuous distribution function with its parameter list as the left argument.   The right operand is a relational function; the right argument is the value of interest.  The syntax for the distribution form of the probability operator is:

          [Parameters]  distributionFunction prob relationalFunction Value

The parameters are optional for the normal and rectangular distributions, defaulting to 0 1.  Some examples follow:

What is the probability of getting exactly two heads when three coins are tossed?   

   3 0.5 binomial prob = 2
0.375

What is the probability of getting at least two heads when three coins are tossed?   

   3 0.5 binomial prob ≥ 2
0.5

What is the probability that a standard normal random variable is less than 1.25?   

   normal prob < 1.25
0.89435

What is the probability that a person is between 70 and 74 inches tall, given that the mean height is 68 inches and the standard deviation is 3 inches?  

   68 3 normal prob between 70 74
0.22974