Experiments with Equally Likely Outcomes: Every experiment has some well-defined set of outcomes. Equally probable denotes that each experiment’s outcome has an equal chance of occurring. If you toss a fair, six-sided die, for example, each of the faces \(1,2,3,4,5,6\) has the same chance of appearing as the others. A fair coin flip will provide a Head \((H)\) or a Tail \((T)\). If you choose a correct or incorrect response to a true or false question on a test at random, you are equally likely to get it right.

In this article, we will discuss the experiments with equally likely outcomes in detail.

Terminologies

Experiment: An experiment is a sort of event in which the outcome is uncertain. Every experiment has a few favourable and negative outcomes.

Scientists will try hundreds of times before making a successful invention.

Random Experiment: An experiment with a known set of possible outcomes is known as a random experiment. Even yet, the exact outcome of a given experiment cannot be anticipated before it is carried out.

Example: Tossing a coin, rolling a die

Trial: The many efforts performed throughout an experiment are referred to as trials. To put it another way, a trial is any specific result of a random experiment.

Tossing a coin is an example.

Event: An event is a trial that has a definite outcome. An event is defined as a result of a coin toss, such as receiving a tail.

Random Event: A random incident is difficult to predict. The probability of such occurrences is extremely low. The appearance of a rainbow in the rain is purely coincidental.

Outcome: A collection of all possible outcomes is known as the outcome of an event. 

Example: When a player strikes a ball towards the goal post, two possible outcomes occur. He has a shot at scoring or missing the goal.

Equally Likely Outcomes: Equally likely outcomes are the results of an experiment in which each of the possible outcomes has the same chance of occurring.

Example: When we roll a die, we get an equal probability of getting any number from \(1\) to \(6\).

\(\text {Probability of any number} =\frac{1}{6}\)

Sample Space: The sample space is the set of results from all the trials in an experiment.

The possible outcomes of rolling dice are \(1,2,3,4,5\), and \(6\).

These results constitute the sample space. \(S = \left\{ {1,\,2,\,3,\,4,\,5,\,6} \right\}\)

Probable event: An occurrence that can be predicted is referred to as a “probable event.” We can assess the likelihood of something like this happening.

Example: The chance of a kid getting promoted to the next class may be predicted, which is a likely event.

Impossible Event: An impossible event does not occur in the sample space of the experiment’s results or is not a part of the experiment.

For example, getting the number \(7\) on the face that is turned up is impossible when a die is thrown.

Complementary Events: When there are only two possible outcomes, one of which is opposite, complementary occurrences occur.

The complement of an event with the probability \(P(E)\) is \(P(\overline{E)}\).

\(P(E)+P(\overline{E)}=1\)

In general, it is true that for an event \(E\),

\(\Rightarrow P(\overline{E)}=1-P(E)\)

Or

\(\Rightarrow P(E)=1-P(\overline{E)}\)

The event \((\overline{E)}\), representing ‘not \(E^{\prime}\) is called the complement of the event \(E\). We can also say that \(E\) and \((\overline{E)}\) are complementary events.

Mutually Exclusive Events: Mutually exclusive events are those in which the occurrence of one prohibits the occurrence of the other. In other words, two occurrences are said to be mutually exclusive if they cannot occur simultaneously.

Tossing a coin, for example, can result in either heads or tails. Viewing both at the same time is impossible.

Equally Likely Outcomes

If all of the outcomes in a sample space have the same likelihood of occurring, they are equally probable. It is difficult to determine if the results are equally probable, but we will assume that the outcomes are equally likely in the majority of the trials. In the following instances, we will use the equally likely assumption. 

(1) Tossing a coin or coins

When a coin is tossed, it can land on one of two sides: heads or tails. If nothing else is said, we will presume that heads and tails are equally likely. If more than one coin is tossed, it will be considered that heads and tails are equally likely.

(2) Throw of a die or dice

A single dice can result in six different results. The probability of each of the six outcomes is considered to be equal. The six faces are believed to be equally likely to appear for any number of dice.

(3) Playing cards

A deck of regular playing cards has \(52\) cards. Because all of the cards have the same size, they are believed to be equally likely to be picked.

(4) Balls from a bag

In probability, there are various scenarios in which certain objects are chosen from a given container. The things in the container are believed to have an equal chance of being selected. A well-known example is the picking of a few balls from a bag of several coloured balls. The balls in the bag are considered to have an equal chance of being chosen.

Not Equally Likely Outcomes

The outcomes of a sample space are not equally likely when they do not all have an equal chance of happening. When a matchbox is tossed, the odds of all six faces appearing are not equal. When a bag of balls of various sizes is opened and chosen at random, all balls are not equally likely to be chosen.

Probability of Equally Likely Outcomes 

If all the outcomes of a sample space are equally likely, then the probability that an event will occur is equal to the ratio: 

\(\frac{\text { The number of outcomes favourable to the event }}{\text { The total number of outcomes of the sample space }}\)

Suppose that an event \(E\) can happen in \(h\) ways out of a total of \(n\) possible equally likely ways. Then the classical probability of occurrence of the event is denoted by

\(P(E)=\frac{h}{n}\)

The probability of non-occurrence of the event \(E\) is denoted by

\(P(\operatorname{not} E)=\frac{n-h}{n}=1-\frac{h}{n}=1-P(E)\)

Thus \(P(E)+P(\operatorname{not} E)=1\)

The event ‘not \(E\)’ is denoted by \(\bar{E}\) or \(E^{\prime}\) (complement of \(E\))

Therefore \(P(E)=1-P(E)\)

Solved Examples: Experiments with Equally Likely Outcomes

Q.1. \(‘A\) and \(B\) are playing tennis,’ says the first question. The outcome of the contest if \(A\) or \(B\) wins is  _____.
Ans: Both \(A\) and \(B\) have an equal probability of winning the game.
As a result, it’s an equally likely event.

Q.2. Assume that each child born has an equal chance of being a male or a girl. Consider the case of a family of three children. List the eight elements in the sample space whose outcomes are all three children’s possible genders.
Answer: The following is a list of all possible genders:
\(S = \left\{ {BBB,~BBG,~BGB,~BGG,~GBB,~GBG,~GGB,~GGG} \right\}\)

Q.3. A standard deck of cards has \(52\) cards that are split into four suits. Diamonds and hearts are represented by red suits, whereas black suits represent clubs and spades. Face cards are the \(J, Q\), and \(K\) playing cards. Let’s say we choose a card at random from the deck. What is the experiment’s sample space?
Ans: In the sample space \(S\), the outcomes are \(52\) cards in the deck.

Q.4. Give three examples of trials with equally likely outcomes.
Ans: Example 1: State true and false:
True or false questions and responses have two equally likely outcomes, i.e. probability to occur true \(=\) probability to occur false \(=\frac{1}{2}\)
Hence it is an equally likely event.
Example 2: Playing a card: Because a deck includes \(52\) cards and the chance of swiping one card of any colour is \(\frac{1}{52}\), playing a card is an equally likely event.
Example 3: Drawing balls from a bag: drawing one of them is an equally likely result if a bag has an equal number of colourful balls.

Q.5. Give any two examples that do not have equally likely outcomes.
Ans:
Example 1: Selecting a ball from a bag containing a variety of sizes and colours.
Example 2: Rolling a Matchbox; Matchboxes do not have equal surfaces on all sides.

Summary

The terms experiment, random experiment, trial, event, random event, outcome, equally likely outcome, sample space, possible event, impossible event, complimentary event, and mutually exclusive event have been discussed in this article. Then we talked about some experiments with equally likely and less likely outcomes, as well as probability calculations. Then we talked about some experiments with equally likely and not equally likely outcomes and calculating the probability of equally likely outcomes.

List of Important Probability Formulas

Frequently Asked Questions (FAQs)

Q.1. How do you know if outcomes are equally likely?
Ans: If each outcome in a sample space \(S\) has the same probability of occurring, the outcomes are equally likely.

Q.2. What is the difference between mutually exclusive and equally likely events?
Ans: Two occurrences that can’t happen at the same moment are said to be mutually exclusive (e.g. going up and down a single elevator simultaneously).
Equally probable occurrences are those that have the same mathematical chance of occurring. A coin flip, for example, has a one-in-two probability of landing on heads.

Q.3. Are the sample space outcomes equally likely?
Ans: The sample space of an experiment is the collection of all possible outcomes. When the likelihood of each event is equal, the results of an experiment are equally likely to occur.

Q.4. What is a list of all possible outcomes called?
Ans: The sample space \((S)\) is a list of all the possible outcomes.

Q.5. How do you describe an outcome?
Ans: The improvements you anticipate seeing as a result of your programme are known as outcomes. These might be improvements you want to see in people, systems, policies, or organisations. Changes in relationships, knowledge, awareness, capacities, attitudes, and actions may be reflected.

We hope this detailed article on experiments with equally likely outcomes helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!