The probabilityeventis a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. The particular output of sample space is called event

A **single outcome** may be an element of many different events, and different events in an experiment are usually not **equally likely**, since they may include very different groups of outcomes. An event defines a **complementary event**, namely the complementary set (the event *not* occurring), and together these define a Bernoulli trial.

Typically, when the **sample space** is finite, any subset of the sample space is an **event **. However, this approach does not work well in cases where the sample space is **uncountably infinite**. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events. [Wikipedia]

**Example:**

**(1)** S={ H , T }

**Possible Events**[latex]= 2^n = 2^2 = 4[/latex]

**Solution:**

**Possible Events**[latex]= 2^n = 2^6 = 64[/latex]

## Types of Probability Events

- Impossible event
- Sure/Certain event
- Equally likely events
- Mutually exclusive event
- Exhaustive Event

## (1) Impossible event

An event that neither occurs is called impossible event

It is denoted by ɸ

Probability of impossible event is zero

P(impossible event) = 0

P( ɸ ) = 0

## (2) Sure/Certain event

An event which surely occur in any condition is called sure or certain event

It is denoted by ‘S’

P(sure event) = 1

P(S) = 1

The range of probability is between 0 & 1

** 0 ≤ P(E) ≤ 1**

## (3) Equally likely events

Two or more events are said to be equally likely events if they have an equal chance of occurrence

**Example:**

S={1, 2, 3, 4, 5, 6}

A={1, 3, 5} B={2, 4, 6}

Above A & B are equally likely event as

A∪B= {1, 2, 3, 4, 5, 6}

## (4) Mutually Exclusive Events:

Two or ore events are said to be mutually exclusive events if they cannot occur simultaneously

**Inset Notation:**

If A and B are two events then A*∩*B= ɸ

**Example:**

S={1, 2, 3, 4, 5, 6}

A={1, 3, 5} B={2, 4, 6}

A∪B= { } = ɸ

## (5) Exhaustive Events

Two or more mutually exclusive events are said to be exhaustive events if they constitute the sample space

**Example:**

# Probability of an event

If E is an event then probability of an event is denoted by P(E) is given as

[latex]P(E) = \frac{Total\: favorable\: cases}{Total\: possible\: cases}[/latex]

OR

[latex]P(E)= \frac{n(E)}{n(S)} = \frac{ No.\: of \: sample \: probability \: in \: E }{ No. \: of \: sample \: probability \:in \: S }[/latex]

**P(E) = ɸ ** **E**= Impossible event , denoted by ‘ɸ’

**P(E) = 1 ** **E**= Sure event denoted by ‘S’

## Example1:

**A card is drawn at random from a pack of 52 plane cards what is the probability that card is**

- Red
- Black
- Diamond
- Pictured Cards
- Divisible by 3
- King
- Red King
- Black Queen
- Jack of club
- Pictured card of heart

**Solution:**

**1.**P(a red card) =[latex]\frac{26}{52}\:= \frac{1}{2}[/latex]

**2.**P(a black card) = [latex]\frac{26}{52}\:= \frac{1}{2} [/latex]

**3.**P(a diamond card) = [latex]\frac{13}{52}\:= \frac{1}{4}[/latex]

**4.**P(a pictured card) = [latex]\frac{12}{52}\:= \frac{3}{13}[/latex]

**5.**P(a card divisible by 3) = [latex]\frac{12}{52}\:= \frac{3}{13} [/latex]

**6.**P(a king) = [latex]\frac{4}{52}\:= \frac{1}{13} [/latex]

**7.**P(a red king) =[latex] \frac{2}{52}\:= \frac{1}{26} [/latex]

**8.**P(black queen) = [latex]\frac{2}{52}\:= \frac{1}{26} [/latex]

**9.**P(a jack of club) = [latex]\frac{1}{52}[/latex]

**10.**P( pictured card of heart) = [latex]\frac{26}{52}[/latex]

## Example 2:

**A coin is tossed 3 times what is the probability the 3 coin shows**

- 1 Heads
- 2 Heads
- 3 Heads
- At-least 1 Head
- At-least 2 Head
- At-least 3 Head
- At-most 1 Head
- At-most 2 Head
- At-most 3 Head

**Solution:**

No. of possible outcomes =[latex] n(S) = 2^3 =8[/latex]

S = { HHH, HHT , HTH , HTT , THH ,THT ,TTH ,TTT }

**1.**[latex]E_1[/latex] = { 1 Head }

={ HHT, THT ,TTH }

[latex]P( E_1) =\frac{n(E_1)}{n(S)}=\: \frac{3}{8}[/latex]

**2.**[latex]E_2[/latex] = { 2 Head }

={ HHT, HTH, THH }

[latex]P( E_2) =\frac{n(E_2)}{n(S)}=\: \frac{3}{8}[/latex]

**3.**[latex]E_3 [/latex] = { 3 Head }

={ HHH }

[latex]P( E_3) =\frac{n(E_3)}{n(S)}=\: \frac{1}{8}[/latex]

**4.**[latex]E_4 [/latex]= { At-least 1 Head }

= { HHH, HHT , HTH , HTT , THH ,THT ,TTH }

[latex]P( E_4) =\frac{n(E_4)}{n(S)}=\: \frac{7}{8}[/latex]

**5.**[latex]E_5 [/latex]= { At-least 2 Head }

= { HHH, HHT , HTH , THH }

[latex]P( E_5) =\frac{n(E_5)}{n(S)}=\: \frac{4}{8} =\: \frac{1}{2} [/latex]

**6.**[latex]E_6 [/latex]= { At-least 3 Head }

= { HHH }

[latex]P( E_6) =\frac{n(E_6)}{n(S)}=\: \frac{1}{8}[/latex]

**7.**[latex]E_7 [/latex]= { At-most 1 Head }

= { HHH, THT, TTH, TTT }

[latex]P( E_7) =\frac{n(E_6)}{n(S)}=\: \frac{4}{8} =\: \frac{1}{2} [/latex]

**8.**[latex]E_8 [/latex]= { At-most 1 Head }

= { HHT , HTH , HTT , THH ,THT ,TTH ,TTT }

[latex]P( E_8) =\frac{n(E_8)}{n(S)}=\: \frac{7}{8}[/latex]

**9.**[latex]E_9 [/latex]= { At-most 3 Head }

= { HHH, HHT , HTH , HTT , THH ,THT ,TTH ,TTT }

[latex]P( E_9) =\frac{n(E_9)}{n(S)}=\: \frac{8}{8}[/latex]