**Random Experiment:**

- If an experiment has more than one possible results which are known in advance and it is not possible to predict which one is going to occur, then the experiment is called a random experiment.
- Examples: Tossing a fair coin, drawing a card from a well-shuffled pack of cards.

**Outcome:**

- The result of a random experiment is called an outcome.

**Sample Space:**

- A sample space of an experiment is the set of all possible distinct outcomes of the experiment and it is denoted by ‘S’.
- Example: A sample space for different events are as follows

**A fair coin is tossed:**

S = { H, T}

∴ Total number of outcomes = 2 ∴ n (S) = 2

**Two fair coins are tossed or a Fair coin is tossed twice:**

S = { H, T} × { H, T} = {HH, HT, TH, TT}

∴ Total number of outcomes = 4 = 2² ∴n (S) = 4

**Three fair coins are tossed or a Fair coin is tossed thrice: **

S = { H, T} × { H, T} × { H, T} = {HH, HT, TH, TT} × { H, T} = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

∴ Total number of outcomes = 8 = 2³ ∴ n (S) = 8

**Notes:**

- An element of a sample space is called a sample point.
- If the number of elements in a sample space is finite then the sample space is called a finite sample space.
- If n coins are tossed then the number of outcomes is 2n.
- A fair coin is tossed twice is equivalent to two fair coins are tossed.
- A fair coin is tossed three times is equivalent to three fair coins are tossed.
- A fair coin is tossed ‘n’ times is equivalent to ‘n’ fair coins are
- A word fair is equivalent to unbiased.

**A fair dice is tossed:**

- S = {1,2,3,4,5,6} ∴ n (S) = 6.

**Two fair dice are tossed**

- S = {1,2,3,4,5,6} × {1,2,3,4,5,6}

S | = | { | (1, 1) | (1, 2) | (1, 3) | (1, 4) | (1, 5) | (1, 6) | |

(2, 1) | (2, 2) | (2, 3) | (2, 4) | (2, 5) | (2, 6) | ||||

(3, 1) | (3, 2) | (3, 3) | (3, 4) | (3, 5) | (3, 6) | ||||

(4, 1) | (4, 2) | (4, 3) | (4, 4) | (4, 5) | (4, 6) | ||||

(5, 1) | (5, 2) | (5, 3) | (5, 4) | (5, 5) | (5, 6) | ||||

(6, 1) | (6, 2) | (6, 3) | (6, 4) | (6, 5) | (6, 6) | } |

∴ n (S) = 36

**Notes:**

- The sum of the two numbers on two dice is called the score on two dice.
- The minimum score on two dice is 2 and the maximum score on two dice is 12.
- The cases favourable to a particular score can be read along the diagonal of that score.

**Event:**

- Any subset of a sample space is called an event. An event is denoted by a capital letter.
- Example – 1: A = an event, that a card selected from a pack of cards is a Diamond.

A Diamond card can be obtained in 13C_{1 }= 3 ways

Number of cases favourable to the event A = 13 ∴ n(A) = 13

- Example – 2: B = an event, that 2 cards selected are consisting of one King and other a Queen.

A King can be obtained in 4C_{1} = 4 ways.

After the selection of a King in any ‘one of four ways,

the selection of a queen can be made in 4C1= 4 ways.

∴ n(B) = 4 x 4 = 16

- Example – 3: C = an event, that a die shows a number greater than 3

∴ C = { 4, 5, 6} ∴n(C) = 3

- Example – 4: D = an event that the score on two dice is 4

∴ D = {(1, 3), (2, 2), (3, 1)} ∴ n(D) = 3

- Example – 5: E = an event that the score on two dice is a prime number i.e. 2,3,5,7,11

∴ E = {(1, 1), (1, 2), (2, 1), (1, 4), (2, 3), (3, 2). (4, 1), (1, 6), (2, 5), (3, 4),

(4, 3), (5, 2), (6, 1), (5, 6), (6, 5)}

∴ n(E) = 15

**Impossible Event: **

- If an event is an empty set, then the event is called an impossible event and is denoted by Φ.
- Example – 1: A = an event that the score on two dice is 15. ∴ A = Impossible event.

- Example – 2: B = an event of having a quadratic equation of three roots. ∴ B = Impossible event.

**Certain Event: **

- If an event is the same as the sample space of the experiment then the event is called a certain event or sure event.
- Example – 1: A = an event that a card selected from a pack of 52 cards is either a red card or a black card.

∴ n(A) = 52 = n(S). and A = S

- Example – 2: B = an event that a die shows a number which is odd or even.

B = {1, 2,3,4,5,6} ∴ n(B) = 6 = n(S) and B = S

**Elementary event:**

- If an event contains only one sample point, then the event is called an elementary event or a simple event.
- Example – 1: A = an event, card selected is a queen of hearts.

A queen of hearts can be obtained in one way.

∴ A is an elementary event. ∴ n(A) = 1.

- Example – 2: B = an event that the score on two dice is 12.

B = {(6,6)} \ n(B) = 1

∴ B is an elementary event.

**The complement of an event A:**

- Let A be an event of a sample space S then the event consisting of all the cases of the samples space which are not favourable to the event is called the complement of the event A and is denoted by A’ or ∴ A’ = {x | x ∈ S, but x ∉ A}
**Examples:**

A = an event that card selected is a spade.

∴ A’ = an event that card selected is not spade

B = an event that the score on the two dice is greater than 4.

∴ B’ = an event that the score on the two dice is less than or equal to 4.

C = an event that, the room is lit.

∴ C’ = an event that the room is not lit.

D = an event that, India wins at least one game.

∴ D’ = an event that, India does not win any game.

E = an event, that the room has at least one fan.

∴ E’ = an event, that the room has no fan.

** ****Notes :**

- If A = an event consisting of at least one then A’ = an event consisting of none.
- If S contains n sample points and A contains m sample points. Then A’ will contain (n – m) sample points.

**Important Results:**

(a) Φ’ = S | (b) S’ = Φ |

(c) A U A’ = S | (d) A ∩ A’ = Φ |

(e) n(A) + n(A’) = n(S) |

#### Probability:

- If A is an event of sample space S’ then the probability of event A denoted by P(A) is defined as

P(A) = n(A)/n(S)

#### Important Relations and Their Proofs:

- Prove that P(Φ) = 0
**Proof :**

Since Φ is an impossible event. Hence Φ is an empty set.

∴ n(Φ) = 0

∴ P(Φ) = n(Φ)/n(S) = 0/n(S) = 0

∴ P(Φ) = 0 is proved

- Prove that P(S) = 1
**Proof :**

Since S is a certain event.

∴ n(Φ) = n(S)

∴ P(S) = n(S)/n(S) = 1

∴ P(S) = 1 is proved

- Prove that 0 < P(A) < 1
**Proof :**

If A is an event of sample space S. Then we have

0 < n(A) < n (S)

Dividing by n(S)

0/n(S) < n(A)/n(S) < n (S)/n(S)

∴ 0 < P(A) < 1 (proved)

- Prove that P(A) = 1 – P(A’)
**Proof :**

If S contains n sample points and A contains m sample points. Then A’ will contain (n – m) sample points.

Where A’ is a complement of the set A.

∴ A U A’ = S and A ∩ A’ = Φ

∴ n(A) + n(A’) = n(S)

Dividing by n(S)

∴ n(A)/n(S) + n(A’)/n(S) = n(S)/n(S)

∴ P(A) + P(A’) = 1

∴ P(A) = 1 – P(A’) is proved

**Explanation of the Phrases :**

**i) odds in favour of an event A and (ii) odds against an event A**

- If x cases are favourable to an event A and y cases are not favourable to the event A then we say odds in favour of A are x : y OR odds against A are y : x,