Coin, Die and Playing cards
(i) Coin : It has two sides, viz. head and tail. If we have more than one coin, coins are regarded as distinct, if not otherwise stated.

(ii) Die : A die has six faces marked 1, 2, 3, 4, 5 and 6. If we have more than one die, all dice are regarded as different, if not otherwise stated.

(iii) Playing cards : Its pack has 52 cards. There are four suits, viz., spade, heart, diamond and club, each having 13 cards. There are two colours red (heart and diamond) and black (spade and club) each having 26 cards. In 13 cards of each suit, there are 3 face cards viz. king, queen and jack. So, there are in all 12 face cards in a pack of playing cards. Also, there are 16 honour cards, 4 of each suit viz., ace, king, queen and jack.

The set of all possible outcomes of an experiment is called the sample space of that experiment. It is usually denoted by S. The elements of S are called sample points and the subset of S is called an event.

Note : Elements of sample space are known as sample points.

If an experiment conducted repeatedly under the identical conditions does not give necessarily the same result, then the experiment is called random experiment. The result of the experiment is called outcome.

Different types of events :

1. Simple event: If an event has only one sample point of the sample space, it is called a simple (or elementary) event.
2. Compound event : When an event is composed of a number of simple events, then it is called a compound event.
3. Null event: An event having no sample point is called null event. It is denoted by Φ. It is also known as impossible event.
4. Sure event: The event which is certain to occur is said to be the sure event.
5. Equally likely events : Events are called equally likely, when we do not expect the happening of one event in preference to the other.
6. Mutually exclusive events: A set of events is said to be mutually exclusive, if the happening of one event A excludes the happening of the other event B, i.e., A ∩ B = Φ
7. Exhaustive events : A set of events is said to be exhaustive, if the performance of the experiment always results in the occurrence of at least one of them.

If A is an event of an experiment whose sample space is S, then its probability P(A) is given by
P(A) = \(\frac{n(\mathrm{~A})}{n(\mathrm{~S})}=\frac{\text { Number of favourable cases }}{\text { Total number of exhaustive cases }}\)

P(A ∪ B) = P(A) + P(B) – P(A ∩ B), where A and B are any two events.

P(A ∪ B) = P(A) + P(B), where A and B are mutually exclusive events.

0 ≤ P(A) ≤ 1, for an event A.

P(\(\overline{\mathrm{A}}\)) = 1 – P(A), where P(\(\overline{\mathrm{A}}\)) denotes the probability of not happening the event A.

P(A ∪ \(\overline{\mathrm{A}}\)) = P(S) = 1.

If the event A implies the event B, then P(A) ≤ P(B).

P(A ∪ B ∪ C) = PC A) + P(B) + P(C) – P(A ∩ B) – P(B ∩ C) – P(C ∩ A) + P(A ∩ B ∩ C).

For any two events A and B,
(i) P(A ∩ \(\overline{\mathrm{B}}\)) = P(A) – P(A ∩ B)
(ii) P(\(\overline{\mathrm{A}}\) ∩ B) = P(B) – P(A ∩ B).

Two events A and B associated with the same random experiment are independent, if and only if, P(AB) = P(A).P(B).

Let p₁ p₂, …, p_n be the probabilities of r independent events A₁ A₂,…, A_n respectively. Then, the probability that at least one of r events happen = 1 – (1 -P₁)(1 – p₂) (1 – p₃)… (1 – P_r)
= \(\frac{\text { No. of favourable cases }}{\text { No. of cases which are not favourable }}\)

Odds in favour of an event
= \(\frac{\text { No. of cases which are not favourable }}{\text { No. of cases which are favourable }}\)