Finite Probability

Finite Probability is a very important concept in discrete mathematics. Before we begin let’s understand some basic terminology, that is important in understanding probability theory.


Basic Terminologies


An experiment is some task you do and get an outcome, possibly from a set of different outcomes. Example, throwing a dice would result in a number between one to six.


Sample Space 

Sample space is all possible outcome an experiment. For example, throw a dice and you get the following outcomes.
Sample Space = { 1, 2, 3, 4, 5, 6 }
You cannot get more than the value of sample space.


An event is a subset of sample space. It means desired outcomes from a set of sample space.
It is denoted in set notations as E ⊆ S. For example if you throw a dice, and a six to appear then it’s an event.
Probability is associated with the event and probability of an event is between 0 and 1. If a probability of an event is 1, then it is a certain event.

Probability of an event is denoted as P(E) and 0 < = P(E) <= 1.

Favorable Outcome

When an event occur then it is a favorable outcome when it happens.

Unfavorable Outcome

When the event does not happen then it is an unfavorable outcome as it did not happen. Sometimes, we also need to find unfavorable outcomes of an event.

Equally likely

An event is equally likely if the probability of each outcome in sample space is equally likely.

For example, when we toss a coin we get either Head or Tail out of two outcomes of the sample space. So there is a 50-50 chance of getting a Head or Tail and both are equally likely events.


Formal Definition of Probability

If S is a finite non-empty sample space of equally likely outcomes and E is an event, that is, a subset of S, then the probability of E is

P(E) = |E| / |S|

Example Problem

Question :

A box contains 3 Red balls and 5 Green balls. What is the probability that a ball picked randomly from the box is Red?

Solution :

Total Number of Balls in Box =  3 + 5 = 8

Sample Space = 8

There are 3 ways to pick Red balls, Therefore, Event E = 3

P(E) = 3/8 

The probability of picking a Red ball from the box is 3/8.


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