# Probability

Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed between zero and one. It helps to predict how likely events are to happen. The meaning of probability is basically the extent to which something is likely to happen. This is the basic theory which is also used in the probability distribution, where you will learn the possibility of outcomes for a random experiment.

For example, when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible (H, T). But if we toss two coins in the air, there could be three possibilities of events to occur, such as both the coins show heads or both shows tails or one shows heads and one tail, i.e. (H,H), (H,T), (T,T).

## Probability in Maths

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 it. Probability can range in between 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. Probability for class 10 is an important topic for the students which explains all the basic concepts of this topic. The probability of all the events in a sample space sums up to 1.

## Formula for Probability

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of outcomes and the total number of outcomes.

 Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes

Sometimes students get mistaken for “favourable outcome” with “desirable outcome”. This is the basic formula. But there are some more formulas for different situations or events.

### Probability Tree

The 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: ## 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 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 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.

Conditional Probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

## Probability Density Function

The Probability Density Function (PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. Probability Density Function explains the normal distribution and how mean and deviation exists. The standard normal distribution is used to create a database or statistics, which are often used in science to represent the real-valued variables, whose distribution are not known.

## Probability Terms and Definition

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 ## Examples of Probability

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, P = 1/6

Question 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, P = Number of Favourable Outcome/Total Number of Outcomes = 12/52= 3/13.