# Random Experiments and Events, Sample Space, Definitions of Various Terms

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## Sample Space:

We throw a die once, what are possible outcomes? Clearly, a die can fall with any of its faces at the top. The number on each of the faces is, therefore, a possible outcome. We write the set of all possible outcomes as,

Again, if we toss a coin, the possible outcomes for this experiment are either a head or a tail. We write the set of all possible outcomes as, .

The set associated with an experiment satisfying the following properties:

Each element of S denotes a possible outcome of the experiment.

Any trial results in an outcome that corresponds to one and only one element of the set is called the sample space of the experiment and the elements are called sample points. Sample space is generally denoted by .

Example:

Consider an experiment of rolling a fair die and then tossing a coin. Write the sample space.

Solution:

In rolling a die possible outcomes are and. On tossing a coin the possible outcomes are either a head or a tail. Let H (head) 0 and T (tail).

## Definitions of Various Terms:

**Event:** Let us consider the example of tossing a coin. In this experiment, we may be interested in ‘getting a head’. Then the outcome ‘head’ is an event.

In an experiment of throwing a die, our interest may be in, ‘getting an even number’. Then the outcomes or constitute the event.

We have seen that an experiment which, though repeated under identical conditions, does not give unique results but may result in any one of the several possible outcomes, which constitute the sample space.

Some outcomes of the sample space satisfy a specified description, which we call an ‘event’.

We often use the capital letters etc. to represent the events.

Example:

Let denote the experiment of tossing three coins at a time. List all possible outcomes and the events that

(i) The number of heads exceeds the number of tails.

(ii) Getting two heads.

Solution:

The sample space is

(Say)

If is the event that the number of heads exceeds the number of tails, and the event getting two heads. Then

And