Random Experiments: Observations, Definitions, and Examples

We may engage in various Random Experiments in our daily lives, sometimes repeating the same behaviours and receiving the same result each time. We use words such as impossible, \(50-50\), probably, and certainly to describe the chance of a phenomenon or a result happening. 

If we toss a coin once, the outcome can be either a head or a tail. Both outcomes are equally possible. And thus, the probability of getting ahead is \(\frac{1}{2}\). Likewise, the probability of getting a tail is \(\frac{1}{2}\). 

Random Experiment in Probability

There are many situations in our daily life where we need to take a chance or risk where a particular event can be easily predicted. For example,

– My calculator will work properly during the next mathematics test.

– The sun will rise in the east tomorrow.

– There will be heavy rain two weeks from now.

– It will snow tomorrow.

– I will visit my grandmother next month.

The branch of mathematics that studies a likelihood or a chance of a phenomenon happening is known as probability. The idea of probability was developed in the \({\rm{1}}{{\rm{7}}^{{\rm{th}}}}\) century. 

An experiment is likely to have more than one possible outcome.

An event is a collection of outcomes from possible outcomes in a random experiment. Now, each of the activities mentioned above fulfils the following \(2\) conditions:

1. We can repeat these activities many times under identical conditions.
2. Because chance plays a part and each conclusion has the same probability of being chosen, we cannot predict the outcome of action ahead of time. As a result of the chance to play a role, an activity is performed. Those events whose outcome can not be predicted beforehand is called a random experiment.

Random Experiment Definition

A random experiment is a process in which the outcome cannot be predicted with certainty in probability. Thus, a random experiment is an experiment whose outcome cannot be predicted precisely in advance, although all possible outcomes are known. An experiment’s result is referred to as an outcome.

An event \(E\) of an experiment is a collection of outcomes. The possible outcomes (=2) when a coin is tossed, i.e., head and tail. When a dice is thrown, the possible outcomes \(=6\), i.e., \(1, 2, 3, 4, 5\), and \(6\). When a card is drawn from a deck of \(52\) cards, the event of drawing a queen consists of a queen of spades, queen of diamonds, queen of hearts, queen of clubs, and many more.

Therefore, we see that an event is part of the possible outcomes. If a random experiment has a finite number of equally likely outcomes, then the probability of an event \(E\) can be expressed as:

\(P(E) = \frac{{n(E)}}{{n(S)}}\) where \(n(E)\) is the number of outcomes favorable to the event \(E\), and \(n(S)\) is the total number of possible outcomes.

Random Experiment Examples

Let us look at some examples of random experiments in probability.

Example 1: Is drawing a card from a well-shuffled deck of cards a random experiment?
Solution: While drawing a card from a well-shuffled deck of cards, the experiment can be repeated as the deck of cards can be shuffled every time before drawing a card. Also, any of the \(52\) cards can be drawn, and hence the outcome is not predictable beforehand. Therefore, this is a random experiment.

Example 3: Selecting a table from \(50\) tables without preference is a random experiment?
Solution: Selecting a table from \(50\) tables is an experiment that can be repeated under identical conditions. As the selection of table is without preference, every table has an equal chance of selection, and thus the outcome is not predictable beforehand. Thus, selecting a table out of \(50\) tables is a random experiment.

Example 2: Multiplying \(2\) and \(6\) on a calculator.
Solution: Although the activity of multiplying \(2\) and \(6\) can be repeated under identical conditions, the outcome is always \(12\). Hence, the activity is not a random experiment.

Random Error Experiment

Random or systematic errors account for all experimental uncertainty. Random errors are stochastic fluctuations (in either direction) in measured data caused by the measuring device’s accuracy limitations. The inability of the experimenter to take the exact measurement in precisely the same way leads to random mistakes. 

On the other hand, systematic errors are repeatable inaccuracies that always go in the same direction. Systematic errors are frequently caused by an issue that continues throughout the experiment.

It’s worth noting that systematic and random errors pertain to issues with taking measurements. 

Errors made in calculations or reading the instrument are not considered in the error analysis.

Solved Examples

Example-1: There are \(20\) seats numbered from \(1\) to \(20\) in a row in a cinema hall. If a seat is selected at random from the row, find the probability that the seat number is
a) A multiple of \(3\)
b) A prime number
Ans: The total possible outcomes consist of \(20\) numbers which are from \(1\) to \(20\).
a) The seat number should be a multiple of \(3\) if it is \(3, 6, 9, 12, 15\), or \(18\).
Thus there are \(6\) multiples of \(3\) from \(1\) to \(20\).
Therefore, the number of favorable outcomes \(=6\)
Probability, \(P\)(the seat number is a multiple of \(3\)) \( = \frac{6}{{20}} = \frac{3}{{10}}\).
b) The prime numbers between \(1\) and \(20\) are \(2, 3, 5, 7, 11, 17\), and \(19\).
Thus, there are \(8\) prime numbers in between \(1\) and \(20\).
Probability, \(P\)(a seat number is a prime number)\( = \frac{8}{{20}} = \frac{2}{5}\)

Example-2: A card is drawn randomly from a deck of \(52\) playing cards. The pack consists of \(4\) suits, and each suit has \(13\) cards. Find the probability that the card drawn is
a) Red
b) A diamond
c) A Picture card
Ans: There are \(13\) cards from each suit are they are Ace,\( 2, 3, 4, 5, 6, 7, 8, 9, 10\), Jack, Queen, and King. 
a) Total red cards in a pack of cards \(=26\) (\(13\) hearts and \(13\) diamonds)
Therefore, probability, \(P\)(Red)=\(\frac{26}{52}=\frac{1}{2}\)
b) There are \(13\) diamond cards in a pack of cards.
Therefore, probability, \(P\)(A diamond card)\( = \frac{{13}}{{52}} = \frac{1}{4}\)
c) The picture card consists of \(4\) kings, \(4\) queens, and \(4\) jacks.
Thus, the total number of picture cards \(=4+4+4=12\)
Therefore, probability, \(P\)(A picture card)\( = \frac{{12}}{{52}} = \frac{3}{{13}}\)

Example-3: There are \(5\) red, \(6\) green, and \(7\) blue balls in a bag. One ball is drawn at random. What is the probability that the ball drawn is a 
a) Red ball
b) Green ball
c) Not a green ball
Ans: The total number of balls in the bag \(=5+6+7=18\)
Therefore, \(P\) (Red ball)\( = \frac{5}{{18}}\)
Therefore, \(P\) (Green ball)\( = \frac{6}{{18}} = \frac{1}{2}\)
Out of \(18\) balls, \(12\) are not green.
Therefore, \(P\) (Not a green ball)\( = \frac{5}{{18}}\).

Example-4: A fair die is rolled. Find the probability of getting 
a) \(6\)
b) An odd number
c) A number less then \(3\)
Ans: In rolling a die, there are \(6\) equally likely outcomes, i.e., \(1, 2, 3, 4, 5\), and \(6\).
a) The event of getting a \(6\) consists of the one outcome \(‘6’\)
Therefore, the probability of getting a \(6\), \(P\)(Getting \(6) = \frac{1}{6}\)
b) There are \(3\) favorable outcomes for the event of getting an odd number. There are \(1, 3\), and \(5\).
Therefore, \(P\) (Getting an odd number)\( = \frac{3}{6} = \frac{1}{2}\)
The favorable outcomes of the event of getting a number less than \(3\) are \(1\) and \(2\).
Therefore, \(P\)(Getting a number less than \(3) = \frac{2}{6} = \frac{1}{3}\)

Example-5: If a number of two digits is made without repetition with the digits \(1, 3, 5.\) Then what is the probability that the number formed is \(35\)?
Ans: The two digits formed with the digits \(1, 3, 5\) without repetition are: \(13, 15, 31, 35, 51, 53\). 
Hence, the total number of outcomes \(=6\)
Out of the six numbers formed, only one number is \(35\).
Therefore, the number of the outcome of the number formed being \(35=1\)
Hence, probability \(= \frac{1}{6}\)

Summary

In this article, we learned about the basics of probability with examples and then learned about the random variable. We learned about random probability in detail with the help of examples.

FAQs

Q.1. What is a random experiment in mathematics?
Ans: A random experiment is a process in which the outcome cannot be predicted with certainty in probability. An experiment’s result is referred to as an outcome. An event \(E\) of an experiment is a collection of outcomes. When a coin is tossed, the possible outcomes \(=2\), i.e., head and tail. When a dice is thrown, the possible outcomes \(=6\).

Q.2. What do you call the collection of all possible outcomes of a random experiment?
Ans: The collection of all the possible outcomes of a random experiment is known as sample space.

Q.3. Give one example which is not an example of a random experiment?
Ans: A stone dropped from a rooftop is not an example of a random experiment as the outcome will always be the same, i.e., the stone will always hit the ground.

Q.4. Define probability.
Ans: The branch of mathematics that studies a likelihood or a chance of a phenomenon to occur is known as probability.

Q.5. What is a random error?
Ans: Random or systematic errors account for all experimental uncertainty. Random errors are stochastic fluctuations (in either direction) in measured data caused by the measuring device’s accuracy limitations.
The inability of the experimenter to take the exact measurement, in the same way, leads to random mistakes.

We hope you find this detailed article on random experiments helpful. If you have any doubts or queries regarding this topic, feel to ask us in the comment section. Happy learning!