Random Experiment - Definition, Random Variable & Sample Space

# Random Experiment

The process of observing something uncertain is known as a random experiment. This is the most basic term used in probability and statistics. The result of a random experiment is an outcome and set of outcomes is sample space. The probability of an event can be calculated using favorable outcomes and sample space. The more number of repetitions are there for an experiment the bigger the size of experiment is. The probability of each output of a random experiment can be calculated if the sample space is known.

## Definition

Random experiment is the process to observe the event having an uncertain outcome. When we toss a coin the outcome is uncertain and hence, it can be termed as a random experiment. The result of a random experiment is known as the outcome and the set of all the possible outcomes of an experiment is known as sample space. If we repeat an experiment n number of times, then each time the experiment is done is known as a trial.

## Random Experiment in Statistics

In statistics, the random experiment is the observation of events having uncertain outcomes. For example, in a bag ten bulbs are there among which few are defective. If you start taking out one bulb at a time and check if it is defective or not, this will be known as a trial. Observations are made from each trial, and in the end a statistical inference is made out of it. The number of repetitions of experiment is known as size of experiment. The statistical inference depends a lot on the size of experiment.

## Random Experiment and Random Variable

A random experiment, as we have already discussed, is a process of observing events with uncertainty whose set of outcomes is known as sample space. A random variable is a function which maps every member of the sample space to a real number. It can be represented as $f:S\rightarrow R$.

The value of random variable is dependent on the output of the experiment. It can be denoted by a capital letter. For example, if a die is rolled the sample space, $S$ = $\{1,\ 2,\ 3,\ 4,\ 5,\ 6\}$. The probability of a $6$ coming in the outcome can be written as $P(X = 6)$. A random variable can be discrete or continuous based on the type of value it can take.

## Random Experiment and Sample Space

The output of a random experiment is known as an outcome or event. The set of all the outcomes of a random experiment is known as sample space. If we toss a coin, it becomes a random experiment as the output is uncertain. The output can be head or tail. Then the sample space becomes $S$ = $\{Head,\ Tail\}$.

## Random Experiment in Probability

Random experiment is defined with its set of outcomes known as sample space. When a random experiment is carried out, the probability of getting a certain output is calculated on the basis of number of favorable outcomes and total number of outcomes. If number of favorable outcomes is $n(E)$ and total number of outcomes is $n(S)$ then the probability of occurrence if getting an output $E$ will be,

$P(E)$ = $\frac{n(E)}{n(S)}$