##### Chapter 6

## Probability

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

#### Introduction

**Probability**is “probably” one of the most routine calculative activities that we all human beings do. What’s the time? What’s the day? When would it be done? When would you be back home? The answer to all of these questions includes the concept of probability. It’s the probable thin line between “what it was” and “what it will be”. Mathematically, it deals with the possibilities of random situations or events. The value of probability ranges between zero to one. It is an essential topic in mathematics because it predicts the possibility of the occurrence of a random event.

As you already know by now that probability is the likelihood of occurrence of an event. However, it is crucial to understand that not all probabilities can be predicted with certainty. It ranges from 0 to 1, 0 being an impossible event, and 1 means a certain event. In sample space, the probability of all events adds up to 1.

Let us consider an example for your better understanding; let’s say we toss a coin, it will either show up as a tail or a head, which means there would be only two certain outcomes (H, T). But what if we toss two or more coins together? The possibility increases. Let us assume two coins; there would be three possible outcomes, i.e., (H, H), (T, T), (H, T).

The formula for probability can be defined as the relationship between several favorable outcomes and the total number of outcomes. The ratio of both is defined as the formula for probability.

The **probability tree** is the visualization of different possible outcomes. The two main positions are the branches and end of the tree. The probability of an event is denoted from the branches and the outcomes from the ends of the tree. The **probability tree** helps to identify the pattern like when to multiply and when to add.

The types of probability are categorized into three parts which are listed as follows:

- Theoretical Probability
- Experimental Probability
- Axiomatic Probability

**Theoretical probability** defines the reasoning behind the probability of an event. It identifies the possible outcomes of an event with reasoning. For example, the probability of getting a spade of ten is 1/52.

The next type is experimental probability, and the basis of **experimental probability** is observations of an experiment. The formula is the same as the original probability formula, i.e., the ratio between the number of favorable outcomes and a total number of outcomes. Let us take an example: in a deck of 52 cards, the probability of getting a ten is 4 divided by the total number of outcomes, i.e., 52. So the experimental probability would be 4/52.

The **axiomatic Probability **is based on the axiom rules. The axiomatic probability approach determines the occurrence or non-occurrence of the events. The chances of an event can be quantified with this approach. The axioms are applied to all sorts of types and events. Kolmogorov introduced the axioms, also called Kolmogorov’s three axioms. The three axioms are:

- The probability of an event E will always be greater than or equal to zero. It can never be less than zero.
- The probability of occurrence of sample space will be 1.
- The probability of an event happening is equal to the sum of the probability of both the events for mutually exclusive events.

At MSVgo, you can learn the concept of Binomial Probability in an articulate step-by-step visualization followed by multiple examples of the topic.

Let us assume that an event E can occur in ‘r’ number of ways and the probability ‘p’ is equally likely. Then the probability of the occurrence of an event would be written as:

P(E) = r/n

The failure or the probability that doesn’t occur would be written as:

P(E’) = (n-r) / n = 1 – (r/n)

E’ is represented as an event that will not occur.

Therefore, it will be

**P(E) + P(E’) = 1**

It depicts the total possibilities of an event that will occur or not occur in an experiment and is equal to 1.

The same theoretical probability of the events is known as **equally likely events**. Event is the single outcome of an experiment, and the outcome is the possible or probable result of an experiment, as you must have noticed in your day-to-day life. The results are equally likely if the occurrence of events all has the same probability. Let us consider an example; if you throw some dice, the probability of getting a 5 is 1/6. Likewise, the probability of getting the remaining numbers one by one would be 1/6. The example of **equally likely events** are as follows:

- Getting a 10 or 2 in a deck of cards
- Getting 1, 4 and 6 on a dice
- Getting a pink colored book from a stack of books

The probability of every event would have the same **equally likely event** probability.

The **complementary events** are the probability, which has only two outcomes. For example, the probability of a person coming to your house would be either yes or no. It is known as **complementary events**. The opposite of a probability is a complement of an event. More examples of **complementary events **are as follows:

- Today, you will either go to a place or not
- You will win the lottery, or you won’t
- You will either pass this exam or not

The probability density function can be defined as the relationship between the density of continuous random variables and ranges of values. It determines and identifies the distribution of the existence of mean and deviation. The standard distribution is used to determine the database and statistics used for the science representation and real-valued variables if the distribution is unknown.

Using the MSVgo app that provides a step by step conceptual breakdown of topics, you can understand the concept of “Additional Rule of Probability” and “Types of Complementary Events of Probability” in a very visually creative manner. To download the app now and see for yourself, click here.