# 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.

## 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 event to occur, how likely they are to happen, using probability.

Probability can range in between 0 to 1, where 0 probability means the event to be an impossible one and probability 1 indicates the 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. Occurrence of any event lies between impossible and certain ( 0 to 1).

### Probability Formula-

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

### Terms Related To Probability

Some of the important 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 Deck of Cards: 4 of hearts is a sample point. the queen of Clubs is a sample point. Experiment or Trial A series of action where the outcomes are always uncertain. 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. Complementary 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

## Types of Probability

There are two major types of probabilities:

Theoretical Probability

Based on the possible chances of something happen. It is based on the reasoning behind probability.

Experimental Probability

Based on the basis of the observations of an experiment. It can be calculated based on the number of possible outcomes by the total number of trials

### 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 final outcome. Tree diagram used to figure out when to multiply and when to add. You can see below a tree diagram for coin:

## Probability Examples

Example 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 event= 4×3 = 12 (considered Jack, Queen and King only)

Probability = $\frac{Number\ of\ favourable\ outcome}{Total\ Number\ of\ outcomes}$

= 12/52