Probability

Probability defines the likelihood of occurrence of an event. There are many

real-life situations in which we may have to predict the outcome of an event. We

may be sure or not sure of the results of an event. In such cases, we say that

there is a probability of this event to occur or not occur. Probability generally has

great applications in games, in business to make predictions, and also it has

extensive applications in this new area of artificial intelligence.

The probability of an event can be calculated by the probability formula by simply

dividing the favourable number of outcomes by the total number of possible

outcomes. The value of the probability of an event happening can lie between 0

and 1 because the favourable number of outcomes can never be more than the

total number of outcomes. Also, the favorable number of outcomes cannot be

negative. Let us discuss the basics of probability in detail in the following

sections.

What is Probability?
Probability can be defined as the ratio of the number of favorable outcomes to

the total number of outcomes of an event. For an experiment having 'n' number

of outcomes, the number of favorable outcomes can be denoted by x. The

formula to calculate the probability of an event is as follows.

Probability(Event) = Favorable Outcomes/Total Outcomes = x/n

Probability is used to predict the outcomes for the tossing of coins, rolling of dice,

or drawing a card from a pack of playing cards. The probability is classified into

two types:

● Theoretical probability

● Experimental probability

To understand each of these types, click on the respective links.

Terminology of Probability Theory

The following terms in probability theorey help in a better understanding of the

concepts of probability.

Experiment: A trial or an operation conducted to produce an outcome is called

an experiment.

Sample Space: All the possible outcomes of an experiment together constitute a

sample space. For example, the sample space of tossing a coin is {head, tail}.

Favorable Outcome: An event that has produced the desired result or expected

event is called a favorable outcome. For example, when we roll two dice, the

possible/favorable outcomes of getting the sum of numbers on the two dice as 4

are (1,3), (2,2), and (3,1).

Trial: A trial denotes doing a random experiment.

Random Experiment: An experiment that has a well-defined set of outcomes is

called a random experiment. For example, when we toss a coin, we know that we

would get ahead or tail, but we are not sure which one will appear.
Event: The total number of outcomes of a random experiment is called an event.

Equally Likely Events: Events that have the same chances or probability of

occurring are called equally likely events. The outcome of one event is

independent of the other. For example, when we toss a coin, there are equal

chances of getting a head or a tail.

Exhaustive Events: When the set of all outcomes of an event is equal to the

sample space, we call it an exhaustive event.

Mutually Exclusive Events: Events that cannot happen simultaneously are called

mutually exclusive events. For example, the climate can be either hot or cold. We

cannot experience the same weather simultaneously.

Events in Probability
In probability theory, an event is a set of outcomes of an experiment or a subset

of the sample space. If P(E) represents the probability of an event E, then, we

have,

● P(E) = 0 if and only if E is an impossible event.

● P(E) = 1 if and only if E is a certain event.

● 0 ≤ P(E) ≤ 1.

Suppose, we are given two events, "A" and "B", then the probability of event A,

P(A) > P(B) if and only if event "A" is more likely to occur than the event "B".

Sample space(S) is the set of all of the possible outcomes of an experiment and

n(S) represents the number of outcomes in the sample space.

P(E) = n(E)/n(S)
P(E’) = (n(S) - n(E))/n(S) = 1 - (n(E)/n(S))

E’ represents that the event will not occur.

Therefore, now we can also conclude that, P(E) + P(E’) = 1

Probability Formula
The probability equation defines the likelihood of the happening of an event. It is

the ratio of favorable outcomes to the total favorable outcomes. The probability

formula can be expressed as,

i.e., P(A) = n(A)/n(S)

where,

● P(A) is the probability of an event 'B'.

● n(A) is the number of favorable outcomes of an event 'B'.

● n(S) is the total number of events occurring in a sample space.

Different Probability Formulas

Probability formula with addition rule: Whenever an event is the union of two

other events, say A and B, then

P(A or B) = P(A) + P(B) - P(A∩B)

P(A ∪ B) = P(A) + P(B) - P(A∩B)

Probability formula with the complementary rule: Whenever an event is the

complement of another event, specifically, if A is an event, then P(not A) = 1 -

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

P(A) + P(A′) = 1.

Probability formula with the conditional rule: When event A is already known to

have occurred, the probability of event B is known as conditional probability and

is given by:

P(B∣A) = P(A∩B)/P(A)

Probability formula with multiplication rule: Whenever an event is the

intersection of two other events, that is, events A and B need to occur

simultaneously. Then

● P(A ∩ B) = P(A)⋅P(B) (in case of independent events)

● P(A∩B) = P(A)⋅P(B∣A) (in case of dependent events)