# Probability Concept

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The probability concept is frequently used in day to day life. In situations like maybe it rains tomorrow, probably I will win this match are some cases of incidents that will happen or not. so the probability is a concept which deals with the measurement of uncertainties.
Here the concept of probability mainly deals with three kinds of situations.
1. Tossing coin (or) rolling die
2. Gambling with cards
3. The number of combinations of objects.
These are the basic concepts in probability around which questions are asked. Let’s understand these three situations…

## Tossing Coin or Rolling Die in Probability Concept

### Coin

A coin consists of two faces Heads and Tails. When coin is tossed events of happening head or tail prevents occurring of event of tails or heads respectively.
so, when a coin is tossed
The total number of outcomes is 2.
The event of the happening of heads or tails is 1.

### Die

A die consists of ‘6’ faces numbered or dotted on each face individually as 1, 2, 3, 4, 5 and 6 (or) with dots.
When a die is thrown the possible number of outcomes is 6.
When two dies are thrown it is 62
For 3 dies it is 63 . Like this, it goes on

## Gambling with Cards in Probability concept

A pack or deck of playing cards contains 52 cards. These 52 cards are categorized into 4 suits.
1. Hearts
2. Diamonds
4. Clubs
And again these 4 suits contain 13 cards individually in each of the suits.
These 13 cards contain one Ace, (2 to 10) numbered cards, one jack, one queen and one king for each suit.

## Number of Combinations of Objects in Probability Concept

Consider ‘n’ objects taken ‘r’ at a time where (r ≤ n) then number of combinations is given by C(n,r) or ncr .
\begin{align*}
n_{C_{r}}=\dfrac {n!}{\left( n-1\right) !r !} \\ \\ = \dfrac {n\left( n-1\right) \left( n-2\right) \ldots \left( n-r+1\right) }{r\left( r-1\right) \left( r-2\right) \ldots 3\cdot 2\cdot 1}
\end{align*}
Here nc0=1    ncn =1

## Some basic terms in Probability Concept

### Trial (or) Experiment

Performing of an experiment is called a Trial.
Example:
1.Tossing a coin
2. Throwing a die

### Event

It is a possible outcome of an experiment or trial.
Example:

• When a coin is tossed, occurring of the head is an event with the head as an outcome.
• Similarly occuring of the tail is also an event with a tail as an outcome.
• When throwing a die possibility of outcomes 1 or 2 or 3 or 4 or 5 or 6 is also an event.

## What is Probability?

Let E be an event
Probability of an event P(E) is defined as number of times event achieved is successfully divided by the total number of possible outcomes.
\begin{align*}
P\left( E\right) =\dfrac {\begin{pmatrix} Number \ of \ times \\ event \ achieved \end{pmatrix}}{\begin{pmatrix} Total \ number \ of \\ possible \ outcomes \end{pmatrix}}\end{align*}

Example:
Let us consider tossing a coin. Find the probability of occurring heads as an event.
Solution:
When a coin is tossed its outcome may have events heads or tails.
so, the total number of possible outcomes =2
heads may occur only once so, number of times event occurs =1
∴ P(E)= ½

### Sample Space

It is the total number of outcomes that occur when a trial is done and is denoted by S.
For tossing a coin S={head, tail}
similarly for die is thrown S={1, 2, 3, 4, 5, 6}
NOTE:

• When a coin is tossed occuring of outcomes is likely to equal in all trials. [heads or trials]
• Similarly when a die is thrown it also has equally likely outcomes.[1, 2, 3, 4, 5&6]

### Simple and Compound Events

In a simple event, we consider the probability of occurring or not occurring in a single event.
Example question of probability on cards
What is the probability of drawing a king spade card from a pack of cards?
Solution:

As we have 52 cards
The total number of possible ways of drawing a card is 52.
Number of ways of drawing King spade card = 1
Probability of event occurring P(E)

\begin{align*}
P\left( E\right) =\dfrac {\begin{pmatrix} Exact \ Number \ of \ times \\ king \ spade \ card \ drawn \end{pmatrix}}{\begin{pmatrix} Total \ number \ of \ cards \\ can \ be \ drawn \ from \ pack \end{pmatrix}}\end{align*}

\begin{align*}= \dfrac{1}{2}
\end{align*}

In compound events, we consider probability of occuring or non occurring of two event or more events at a time.
Example Probability question on balls
If fro a bag containing 8 blue &5 green balls two successive draws of 2 balls are made without replacement. Then what is the probability of getting 2 blue balls in first draw and 2 green balls in the second draw?
Solution:

Since the bag contains 8 blue balls & 5 green balls
Total number of balls in bag =13
The favorable case of the first draw is 2 balls from 8 balls.
Probability in the first case ( P1 )= 8c2 / 13c2
After this incident the total number of balls in the bag are 13-2=11
In the second case drawing of 2 green balls from 5 green balls is made.
Probability in the second case ( P2 )= 5c2 / 11c2
Both events are independent therefore total probability= P1 × P2
⇒ ( 8c2 / 13c2 ) × ( 5c2 / 11c2 )

### Exclusive Events

If the occurrence of one event resists or rules out the occurrence of another event in the same experiment is a mutually exclusive event.
Example:
When a coin is tossed events of heads resists event of tails. Here both head & tail cannot occur at the same time. so, they are mutually exclusive events.

### Independent Events

Two events A and B are said to be independent events if the probability of occurring of one event (A) does not affect probability of occurrence of another event (B).
Example:
When two coins are tossed simultaneously so the occurrence of tails on the first coin does not affect the occurance of heads on another coin.
Thus, they are independent events.
Probability of two independent events is P(A & B)=P(A) × P(B)

## Algebra of Events in Probability Concept

Consider A and Y as events within sample space (S) then,

 Algebra Notation Probability Representation A∪Y Either event A or Y or both happen A∩Y Event A and Y both happen simultaneously Ā Event A does not happen Ȳ Event Y doesnot happen Ā∩Ȳ Event of non occurance of both A and Y

## Rules of Probability

In a random experiment let E and F be the events and ‘S’ be sample space.

1. Probability of an event is always considered to be lie between 0 and 1. (0≤P(E)≤1)
2. Probability of sum of all the elementary events of a trial is always equal to 1.
3. Probability of occurance of an impossible event is ‘0’.
Example:

When a die is thrown its outcomes are {1, 2, 3, 4, 5 and 6}
Event of occurance of ‘8’ is not possible.
so, probability of occurance of event ‘8’ is impossible when a die is thrown.

4. For sure event probability is ‘1’.
Example:

When a die is thrown its outcomes are {1, 2, 3, 4, 5 and 6}
For a die thrown event of happening a number less thean ‘7’ can be achieved in 6 ways.
Its total outcomes are 6
∴ Probability of occurance of event less than 7 = 6/6 =1
Thus, probability of a event that is sure to occur is ‘1’.

5. P(Ē)=1-P(E)
Here Ē is complementary event of ‘E’.
6. P(E∪F)=P(E)+P(F)-P(E∩F)
7. P(E-F)=P(E)-P(E∩F)