Introduction to Probability

Probability / Saturday, June 23rd, 2018

Introduction to Probability

The word Probability is synonymous with the word ‘chance’. We use the word ‘probability’ in many situations with out understanding a definite meaning of the term. For example, we say ‘the probability of getting a head in tossing a coin is 60%, ‘the probability of getting face value 5 in throwing a dice is 30%’ etc.

Here we will try to learn ‘probability’ in mathematical way.

So, lets start by understanding some important terms associated with Probability.

If we already know what are the outcome will be for an experiment but we can’t tell definitely which one will occur at the very moment then that experiment is called ‘random experiment’.

For example, if we toss an unbiased coin, we know that there is two probable outcomes either ‘head’ or ‘tail’. But we can’t definitely tell which one we will get when we actually do the experiment.

In a random experiment all the possible outcomes individually is called ‘event’.

For example, in tossing a coin getting ‘head’ or ‘tail’ is event.

Events are of two types

1. Simple or Elementary
2. Compound

The events which can be further decomposed into simple event is called ‘compound event’ and which are not is called ‘simple event’

For example, in throwing a dice the event of getting ‘even face’ that is either 2 or 4 or 6 is called compound event but only getting face 2 is called simple event.

If two or more events are related to each other in such a way that they can’t occur at the same time is called ‘mutually exclusive events’.

Symbolically let A and B are two events such that

$A \cap B = \phi$

then A and B are mutually exclusive. For example, in throwing a dice getting the event ‘even face’ and ‘odd face’ are mutually exclusive.

The negative event of a specific event is called complementary event. In other words alternative event of a specific event is complementary event.

For example, the event ‘not getting a head’ is complementary event of ‘getting a head’ in tossing a coin.

If A be an event then A’ or Ac is complementary event.

If the probability of all the events of a corresponding random experiment are equal then the events are called equally probable events.

Let A and B are two events of a random experiment if

$P(A) = P(B)$

then A and B are equally probable.

In a random experiment, there is two or more events such that at least one of them will occur at the time of experiment then the events are called exhaustive events.

For example, in tossing a unbiased coin let event A = Getting Head and B = Getting Tail then we will say A and B are exhaustive as one of the certainly occur at the time of toss.

In a random experiment, if there is an event which is not possible to occur in any circumstances is called impossible event. And is denote as  $\phi$ and

$P(\phi) = 0$

In a random experiment, if there is an event which will occur every time then the event is called certain event.

Let S be the certain event of a random experiment then,

$P(S) = 1$

In a random experiment all the simple event is termed also as sample point and the set of all the sample point of the random experiment is called sample space and it is denote with ‘S’.

For example, sample space of throwing dice experiment is,

$S = \{ 1, 2, 3, 4, 5, 6\}$

At the beginning of the 19th century, Laplace gave the formal definition of probability which goes by the name of the classical definition.

Let S be the sample space of a given experiment E which is finite. If all the simple events connected to E be ‘equally likely’ then the probability of an event A is defined as

$P(A) = \cfrac{n(A)}{n(S)}$

where n(A) is the total number of elements in A and n(S) is the total number of elements in the sample space.

$P(\phi) = 0$

$P(A^c) = 1 - P(A)$

$0 \leq P(A) \leq 1$

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

[ where event A and B are not mutually exclusive]

$P(A \cup B) = P(A) + P(B)$

[ where event A and B are mutually exclusive]

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