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Random Experiments

We may perform various activities in our daily existence, sometimes repeating the same actions though we get the same result every time. Suppose, in mathematics, we can directly say that the sum of all interior angles of a given quadrilateral is 360 degrees, even if we don’t know the type of quadrilateral and the measure of each internal angle. Also, we might perform several experimental activities, where the result may or may not be the same even when they are repeated under the same conditions. For example, when we toss a coin, it may turn up a tail or a head, but we are unsure which results will be obtained. These types of experiments are called random experiments.

Random Experiment in Probability

An activity that produces a result or an outcome is called an experiment. It is an element of uncertainty as to which one of these occurs when we perform an activity or experiment. Usually, we may get a different number of outcomes from an experiment. However, when an experiment satisfies the following two conditions, it is called a random experiment.

(i) It has more than one possible outcome.

(ii) It is not possible to predict the outcome in advance.

Let’s have a look at the terms involved in random experiments which we use frequently in probability theory. Also, these terms are used to describe whether an experiment is random or not.

Terms Meaning
Outcome A possible result of a random experiment is called its outcome.

Example: In an experiment of throwing a die, the outcomes are 1, 2, 3, 4, 5, or 6

Sample space The set of all possible outcomes of a random experiment is called the sample space connected with that experiment and is denoted by the symbol S.

Example: In an experiment of throwing a die, sample space is S = {1, 2, 3, 4, 5, 6}

Sample point Each element of the sample space is called a sample point.

Or

Each outcome of the random experiment is also called a sample point.

Learn more about sample space here.

What is a Random Experiment?

Based on the definition of random experiment we can identify whether the given experiment is random or not. Go through the examples to understand what is a random experiment and what is not a random experiment.

Example 1:

Is picking a card from a well-shuffled deck of cards a random experiment?

Solution:

We know that a deck contains 52 cards, and each of these cards has an equal chance to be selected.

(i) The experiment can be repeated since we can shuffle the deck of cards every time before picking a card and there are 52 possible outcomes.

(ii) It is possible to pick any of the 52 cards, and hence the outcome is not predictable before.

Thus, the given activity satisfies the two conditions of being a random experiment.

Hence, this is a random experiment.

Example 2:

Consider the experiment of dividing 36 by 4 using a calculator. Check whether it is a random experiment or not.

Solution:

(i) This activity can be repeated under identical conditions though it has only one possible result.

(ii) The outcome is always 9, which means we can predict the outcome each time we repeat the operation.

Hence, the given activity is not a random experiment.

Examples of Random Experiments

Below are the examples of random experiments and the corresponding sample space.

  1. Tossing a coin three times

    Number of possible outcomes = 8

    Sample space = S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

  2. Three coins are tossed simultaneously

    Number of possible outcomes = 8

    Sample space = S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

  3. Rolling a pair of dice simultaneously

    Number of possible outcomes = 36

    Sample space = S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

  4. Throwing a die two times

    Number of possible outcomes = 36

    Sample space = S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

  5. Selecting a card from an urn containing 100 cards numbering from 1 to 100

    Number of possible outcomes = 100

    Sample space = S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….., 51, 52, 53, 54, 55, …., 91, 92, 93, 94, 95, 96, 97, 98, 99, 100}

  6. Choosing one of the factors of 180

    Number of possible outcomes = 18

    Sample space = S = {1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180}

Similarly, we can write several examples which can be treated as random experiments.

Playing Cards

Probability theory is the systematic consideration of outcomes of a random experiment. As defined above, some of the experiments include rolling a die, tossing coins, and so on. There is another experiment of playing cards. Here, a deck of cards is considered as the sample space. For example, picking a black card from a well-shuffled deck is also considered an event of the experiment, where shuffling cards is treater as the experiment of probability.

A deck contains 52 cards, 26 are black, and 16 are red.

However, these playing cards are classified into 4 suits, namely Spades, Hearts, Diamonds, and Clubs. Each of these four suits contains 13 cards.

We can also classify the playing cards into 3 categories as:

Aces: A deck contains 4 Aces, of which 1 of every suit. 

Face cards: Kings, Queens, and Jacks in all four suits, also known as court cards.

Number cards: All cards from 2 to 10 in any suit are called the number cards. 

  • Spades and Clubs are black cards, whereas Hearts and Diamonds are red.
  • 13 cards of each suit = 1 Ace + 3 face cards + 9 number cards
  • The probability of drawing any card will always lie between 0 and 1.
  • The number of spades, hearts, diamonds, and clubs is the same in every pack of 52 playing cards.

An example problem on picking a card from a deck is given above.

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