# Probability And Random Events

Probability

Introduction:

in daily life we come across many different events. We are interested in finding the chance of occurrence of these events. This is the basis of the concept of probability. Before we understand what is probability, let us understand what are events and the different kinds of events.

Random events: the events whose outcome is unknown are called random experiments. For example when we toss a coin, we do not know if it will land heads up or tails up. Hence tossing a coin is a random experiment. Another example is the result of an interview or examination.

When we speak about random experiments, we have to know what is the sample space.

Sample space denoted by S is the set of all possible outcomes of a random experiment.

Example: consider the random experiment of tossing a die. Let us write down the sample space S here.

The sample space is all the possible outcomes here. What are the possible outocmes when we toss a die once? As we know a die has 6 faces numbered 1,2,3,4,5,6. When we toss it once, only one of the face will turn up. Hence the sample space is

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

Consider one more simple example of tossing two coins. Let us write down the sample space here.

S={(H,H),(H,T),(T,H),(T,T)}

T:tail

Can we think of any example which is non-random? That is whose outcome can be predetermined? Consider the example of executing a computer program. If you write the program correctly then we will know what the outcome of the program will be.

There are different kinds of random experiments. Let us discuss them here.

Event: A subset of a sample space is called as an event. Events are always represented by capital letters A,B,C,etc.

Example: consider the random experiment of throwing a die. The sample space here is

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

Consider an event

A: event of selecting an odd number.

Then the different outcomes of the event A is

A={1,3,5}.

The set A as we see is an subset of the sample space S.

Complement of events: the non-occurrence of any event is called its complement. The complement of any event A is represented by Ac

Example: consider the random experiment of attending an interview. Then the sample space is

S={pass, fail}

That is the two possible outcomes of the interview.

Suppose event A is that the person passes the interview.

Then A={pass}

But if the person fails in the interview, then that is the non-occurrence of event A,which is nothing but its complement.

Ac={fail}.

Independent events: consider any two events A and B. they are said to be independent if the occurrence of one event does not dependent on the occurrence.

For example consider the random experiment of tossing a coin. Then the sample space here is

Let event A:getting a head

Let event B: getting a tail

B={tail}.

Here both the events A and B are independent. This is because when we toss a coin for the first time, suppose we get a head. Again when we toss the coin for the second time, it is the outcome does not depend on the previous outcome. Hence the two events A and B are independent.

Consider another experiment, of birth of a child.

The sample space here is

S={boy, girl}

Suppose event A: a son is born

A={boy}

And event B: a daughter is born

B={girl}.

Here also we see that both the events A and B are independent. This is because the birth of a son in the first delivery does not necessarily ensure that the second child born will be a daughter. Hence both the events A and B are independent.

Mutually exclusive events: any toe events A and B are said to be mutually exclusive, if the occurrence of events A and B is a null set.

Example: consider the random experiment of throwing a die. The sample space here is

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

Consider event A:getting an even number

A={2,4,6}

Event B: getting an odd number

B={1,3,5}

Now let us look at the occurrence of events A and B together, that is we need to look at the common elements in events A and B. as we see here in both the events A and B there is no common element at all. Hence both the events A and B are mutually exclusive events.

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