# Probability

Probability is a branch of mathematics that deals with the occurrence of a random event. The value is expressed between zero and one. Probability helps to predict how likely events are to happen. For example, when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible.

Contents in Probability:

## What is Probability?

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using probability. Probability can range in between 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.

 Note: The probability of all the events in a sample space sums up to 1.

## The Line of Probability

Probability can also be shown on a line. The occurrence of any event lies between impossible and certain (0 to 1).

Line of Probability

### Probability Formula

 Probability of event to happen P(E) = $\frac{Number \; of \; favourable \; outcome}{Total \; Number \; of \; outcomes}$

Some of the important probability terms are discussed here:

Term Definition Example
Sample Space The set of all the possible outcomes to occur in any trial
1. Tossing a coin, Sample Space (S) = {H,T}
2. Rolling a die, Sample Space (S) = {1,2,3,4,5,6}
Sample Point It is one of the possible results In a deck of Cards:

• 4 of hearts is a sample point.
• the queen of Clubs is a sample point.
Experiment or Trial A series of actions where the outcomes are always uncertain. The tossing of a coin, Selecting a card from a deck of cards, throwing a dice.
Event It is a single outcome of an experiment. Getting a Heads while tossing a coin is an event.
Outcome Possible result of a trial/experiment T (tail) is a possible outcome when a coin is tossed.
Complimentary event The non-happening events. The complement of an event A is the event not A (or A’) Standard 52-card deck, A = Draw a heart, then A’ = Don’t draw a heart
Impossible Event The event cannot happen In tossing a coin, impossible to get both head and tail

Events in Probability

## Types of Probability

There are three major types of probabilities:

• Theoretical Probability
• Experimental Probability
• Axiomatic Probability

### Theoretical Probability

It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting a head will be ½.

### Experimental Probability

It is based on the basis of the observations of an experiment. The experimental probability can be calculated based on the number of possible outcomes by the total number of trials. For example, if a coin is tossed 10 times and heads is recorded 6 times then, the experimental probability for heads is 6/10 or, 3/5.

### Axiomatic Probability

In axiomatic probability, a set of rules or axioms are set which applies to all probability types. These axioms are set by Kolmogorov and are known as Kolmogorov’s three axioms. With the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified. The axiomatic probability lesson covers this concept in detail with Kolmogorov’s three rules (axioms) along with various examples.

## Probability Tree

Probability tree diagram helps to organize and visualize the different possible outcomes. Branches and ends of the tree are two main positions. Probability of each branch is written on the branch, whereas the ends are containing the final outcome. Tree diagram used to figure out when to multiply and when to add. You can see below a tree diagram for the coin:

Probability Tree

### Example Questions Based on Probability Concepts

Question 1: Find the probability of rolling a ‘3 with a die.’

Solution:

Sample Space = {1, 2, 3, 4, 5, 6}

Number of favourable event = 1

Total number of outcomes = 6

Thus, Probability = 1/6

Example 2: Draw a random card from a pack of cards. What is the probability that the card drawn is a face card?

Solution:

A standard deck has 52 cards.

Total number of outcomes = 52

Number of favourable events = 4 x 3 = 12 (considered Jack, Queen and King only)

Probability = Number of Favourable Outcome/Total Number of Outcomes = 12/52= 3/13.

#### Practise This Question

Three boys and two girls stand in a queue. The probability that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is?