Contents

**Outcome|Definition & Meaning**

**Definition**

According to **theoretical probability**, the **result** of an **experiment** (could be a trial) is **known** as an** outcome** that **holds the uniqueness** for a **specific experiment**, and also there is **nothing common** **between** different **outcomes** which **makes **them **mutually exclusive**.

In other words, we can say that an outcome is the** acquired output** from performing an **experiment**.

**Conceptual Overview**

Before we dig deeper into the term outcome we have to **refine** our **concept** related to** probability** so **we will revisit** some **terms** as described below. The **illustration** of the **outcome** of **tossing** a fair **coin** is **shown** in the figure.

Figure 1 – Outcome of Tossing a Fair Coin

**Experiment**

When a **set of actions** is **performed** to **produce** an **operation** that **contains** **outcomes** that are **not certain**. For instance, **rolling the die**, **tossing a coin**, and many more.

**Sample Space**

Suppose **we** have **done** a simple **experiment** and we** get** an **outcome** and on **repeating** up to **n possible time**, all the **outcomes** are **represented** in a **space** known as **sample space** that has **all the outcomes** of the **experiment**.

For instance, suppose we **roll a die. **Whatever number comes facing up is an **outcome** of this experiment. However, a die has six faces, and each face has a unique number associated with it as 1, 2, 3, 4, 5, and 6. So the total set of possible outcomes for each experiment is the sample space S = {1, 2, 3, 4, 5, 6}. The “{}” emphasize the sample space’s set-like nature.

**Event **

When an **experiment** is **performed** then the **outcomes** of that **experiment** are **known** as an **event**, in other words, the **specific set** of **outcomes** when an experiment is performed. For instance, if a **die is rolled** the sample space will be **S={1,2,3,4,5,6}** but the **event** may be **defined** as** E={2,4,6}** **set of even number** outcomes, or **E={1,3,5}** which is a **number is odd,** so this represents the event.

**Elementary Event**

An **event** that has a **single outcome** is known as an elementary event. For instance,** tossing a coin** and **getting a head**.

Figure 2 – Conceptual Overview of Outcome of Experiment

**Illustration of Outcomes in Various Experiments**

**Experiment 1**

Suppose we **perform an experiment** in which we** toss a coin** **two times**. Then the **possible outcomes** will be **four**. Making sample space for this experiment gives **S={(H, T), (T, T), (T, H), (H, H)}** Where **H ****and T** represent **head and tail** respectively.

In order to **not mix the event** and outcome terms, we will describe a **small difference** where **events** are a** subset** of the **sample space** like an event having **“two heads when a coin is tossed two times”**. So this event will be equal to** E=(H, H)** while the **outcome** can be **calculated** by $2^{n}$ where **n** is the** frequency of operation. **

In this experiment, a coin is tossed **two times,** so the **frequency **is **2. **Since $2^{2}=4$, there are** four possible outcomes**. This illustration is shown below.

Figure 3 – Illustration of tossing a coin two time

**Experiment 2**

Suppose we have **two dice** and we **roll them** as we know the **outcome** for **one dice** is **6** so** for two dice** there will be **6 x 6 = 36 outcomes**, we can **define** an **event** here as the **outcomes** having the **sum greater than 3**. This illustration is shown in the figure below.

Figure 4 – Illustration of rolling a dice twice and checking outcomes

**Outcome Probability**

**Discrete Domain**

For a** deterministic domain** where is **no infinite sample space**, the **probability** of each **outcome** can be **easily assigned** according to the **given sample space**.

For example, when a **fair** **die is rolled once**, the **sample space is {1,2,3,4,5,6},** and we can write the **probability of individual outcomes **as below.

P(X) = No. of Favourable Outcomes / Total No. of Outcomes

probability that the number 1 faces up = P(1) = 1/6

probability that the number 2 faces up = P(2) = 1/6

And similarly, we get the same probabilities for the rest of the outcomes:

P(3) = 1/6

P(4) = 1/6

P(5) = 1/6

P(6) = 1/6

It is possible for outcomes to have **different probabilities** of occurrence, such as one side of the die being heavier than the rest, etc. Experiments with such biased devices are called **biased experiments**.

**Continuous Domain**

When we are talking in the **continuous domain** where our **sample space** is generally **infinite** then the **probability** for **each outcome** will be **zero** because in this domain **we assign** **probability** in **piecewise ranges** as shown below. The region has** two outcomes X>0.5 and X<0.5** so we assign a probability of **0.5 to one region** and **0.5 to another region** **sums** up giving the **probability of 1**.

Figure 5 – Outcome in Continuous Domain

**Equally Likely Outcomes**

The** probability** of **each outcome** in the **space is equal**, so we can **assume** **all outcomes** are** equally likely** (assume they **occur at equal rates**).

An **ordinary coin**, for instance, is assumed to **have** the **same probability** of landing **“head” or “tail”**. Common **games of chance** use **randomization tools** based on the **assumption** that all **outcomes** have **an equal probability.**

**Various probabilities** are sometimes **treated** as if they are a**ll equally likely**, which is **not the case**. A set of equally likely outcomes c**an’t be used** to **describe all experiments**. For instance,** throwing a thumbtack** and **watching** whether it **lands upward or downward** **doesn’t imply** that both outcomes are **equally likely**.

**Tree Diagram**

In probability, a **tree diagram** helps to **determine the potential outcomes** or likelihoods of an event along with the information that whether the **event occurred or not**. Below is the **illustration** of the tree diagram **where a coin is tossed** and the** probability** of **each outcome** is **shown** below. We can see that the sample space is **S={Head, Tail}** and the **probability** of **each outcome** is** 0.5**.

Figure 6 – Tree Diagram for Probability of an outcome

**Example Problem – Determining Outcomes of an Experiment**

For the given experiment, determine each experiment’s** outcomes and sample space**.

**Experiment 1**: Select a ball from a bag containing **balls** numbered **1 to 20** and note the number of balls.

**Experiment 2**: Tossing a coin **three times** and **noting** the **order of head and tail**.

**Experiment 3**: Tossing a coin **three times** and noting the **number of heads**.

**Experiment 4**: Suppose we have a bag containing **two red balls** numbered **1,2** and **two blue balls** numbered** 3,4** color, and the number of balls is noted.

**Solution**

**Experiment 1**

The **outcome** will be **any ball numbered** from **1 to 20**.

**Sample space** will be all the outcomes from 1 to 20 so:

**S={1, 2, 3, …, 20}.**

**Experiment 2**

The outcome will be **any sequence** from **HHH, HHT, HTH, THH, THT, THH, TTH, and TTT.**

Sample space will be all the outcomes:

**S={HHH, HHT, HTH, THH, THT, THH, TTH, TTT}.**

**Experiment 3**

The outcome will be **any number** of heads from** 0,1, 2, 3.**

Sample space will be all the outcomes:

** S={0, 1, 2, 3}**

**Experiment 4**

The outcome will be **any sequence** from **(1, R), (2, R), (3, B), (4, B).**

Sample space will be all the outcomes:

**S={(1, R), (2, R), (3, B), (4, B)}**

*All mathematical drawings and images were created with GeoGebra.*