# Probability Notation

Sub Topics
A notation is short form by means of a letter, a symbol, or an abbreviation of words for a physical definition. Many times letters from other languages are used to define something. In some cases such notations becomes very unique to the extent that they are understood better than the actual verbatim. For, example the notation ‘∑’ is understood even better than writing the actual word ‘summation’. Notation helps to express the mathematical relations in a simple and condensed manner. As usual, the nomenclatures used in set theory are also used as notations in many topics. In the subject of probability, notations are widely used and let us take a closer look on that.

## Probability Notation Symbols

Let us first take a stock of some important symbols that are seen in the topic of probability.

1)    {a, b, c, d, e, …..}
2)    S
3)    Φ
4)    A
5)    A’
6)    P(A)
7)    P(A’)
8)    A U B
9)    A∩B
10)  P(A U B)
11)  P(A ∩ B)

## Probability Notation Definition

Probability notation definition is nothing but the meaning of the various notations we use. Let us go through some of the definitions with examples.
1)    A set of outcomes of a probability experiment is denoted in a set notation, each item of the set represents an outcome. The notation means the different outcomes a, b, c, ……….., for a particular probability experiment. For example, all possible outcomes of rolling a die are  denoted in a single set as {1, 2 , 3, 4, 5, 6}.

2)    S is a notation that is used for denoting a sample space which is same as the total number of  outcome. In case of rolling a die for any number, S = {1, 2 , 3, 4, 5, 6}.

3)    The Greek letter Φ denotes a empty set. That is when no outcome is possible or an impossible event.

4)    Generally the capital letter A denotes an event. It is defined by a set of outcomes that can exhaust the event. For example, if event A is for rolling even numbers on single die is denoted as A = {2, 4, 6}.

5)    The notation A’ represents an event compliment to A, that is the set which does not occur. For example, if A = {2, 4, 6}, then A' = {1, 2, 3}.

6)    P(A) is the notation for the probability of the event A to occur. In the same example of getting a even number, P(A) = $\frac{{2, 4, 6}}{{1, 2 , 3, 4, 5, 6}}$ = $\frac{1}{2}$

7)    P(A’) is the notation for the probability of the event A not occurring.

8)    A U B is a notation that literally means A union B. In probability it means that event A or event B or both events occur.

9)    A ∩ B is a notation that literally means intersection B. In probability it means that event A and event B occur together.

10)    P(A U B) is the probability when the event A or event B or both events occur.

11)    P(A ∩ B) is the probability when the event A and event B occur together.
At this juncture we will explain some formulas that engage probability notation.

Two events A and B are said to be mutually exclusive if A and B cannot occur at the same time. Therefore, in such a case P(A ∩ B) =  0 or = Φ, an empty set.
At this juncture we will explain some formulas that engage probability notation.
1)    P(A U B) = P(A) + P(B) - P(A ∩ B)

2)    In case of mutually exclusive events, P(A U B) = P(A) + P(B), since P(A ∩ B) = 0 in this case.
Probability Notation Given,

Sometimes an event may be bifurcated into two events which are successive. We may have to calculate the probability of the second event, considering the probability of the first. This is called probability of event B after event A is occurred. It is denoted as P(B/A). The probability formula is,
P(B/A) = [P(A ∩ B)]/P(A)

Let us explain this with an example.

In a competition 40% of the candidates cleared the first interview and 25% of the candidates cleared the second. What is the probability for a candidate to clear the second interview if he has passed the first interview?

The 25% of the candidates come from the lot of candidates who cleared the first one, that is 40 % of the total candidates. That is, P(A ∩ B) = 0.25 and P(A) = 0.4

Therefore,
P(B/A) = P(A ∩ B)/P(A) = 0.25/0.4 ≈ 0.67

Thus, the probability for a candidate who cleared the first interview has about 67% probability to clear the second interview.

## Statistics Probability Notation

Statistics is the base subject that is used in advanced topics of probability. To evaluate a higher level of probability, you require a number of statistical data and hence the notation used in statistics are also considered as probability notation in statistics. The probability notation in statistics are generally as follows.
1)     ẋ for mean, that is, (x1 + x2 + …… + xn)/(n)

2)    s2 for variance

3)    ‘r’ for correlation coefficient

4)    Cumulative sample as kr

5)    ‘s’ for standard deviation.