# Probability Problems, Shortcut Tricks and Examples

**Probability Examples - Page 2**

**Probability Questions - Page 3**

**Probability Lectures - Page 4**

** Probability **In an experiment if ‘ n ‘ is the number of exhaustive cases and ‘m’ is the number of favourable cases of an event A. Then the probability of event

**A**is denoted by

**P(A)**.

**P(A) = **

## Probability Theory

**Random Experiment**: An experiment in which all possible out comes are known and the exact output cannot be predicated in advance is called Random Experiment.

**EXAMPLE :**

(i) Tossing a fair coin

(ii) Rolling an unbiased dice

(iii) Drawing a card from a pack of well shuffled cards**Trail :**Conducting a Random Experiment once is known as a Trail**Outcome**: The result of a Trail in the random experiment is called as Outcome

**EXAMPLE :**In tossing a single coin outcomes are**H**and**T****Sample Space :**A set of all possible outcomes of a random experiments is known as Sample Space

**EXAMPLE :**

( i ) In the experiment of tossing a coin the sample space**S H,T**( ii ) If Two coins are tossed then

**S HH,HT,TH,TT**( iii ) In throwing a die S

**1,2,3,4,5,6****Event :**Any non-empty subset of a Sample Space is called an Event

**EXAMPLE :**( i ). in Tossing a single coin getting Head or Tail is an Event

( ii )Getting an Ace (or) Diamond from a pack of 52 cards is an event**Exhaustive Events :**The total number of possible outcomes of an experiment is known as Exhaustive Events

**EXAMPLE :**In the experiment of throwing a die the total number of possible outcomes = 6**Favourable Events :**The number of event which favour the happing of the events are known as Favourable Cases (or) Events

**EXAMPLE :**In tossing two dice the number of cases favourable to getting the sum 3 is (2, 1), (1, 2) i.e., 2**Mutually Exclusive Events :**If two events have no common outcomes then they are called Mutually Exclusive

**EXAMPLE:**In tossing a coin the events Head & Tail are mutually exclusive because Head & Tail cannot happen at the same time.**Independent Events :**Two events are said to be independent if the happening of one event does not affect the happening of the other.**Dependent Events**: Two events are said to be dependent if the happening of an event will affect the happening of the other event .

**EXAMPLE:**If we draw a card from a pack of 52 cards and replace it before we draw a second card. The second card is independent of first one. If we don’t replace the first card before the second draw. The second draw depends on the first one.**Axioms of Probability :**Let S be the sample space and A be the event i.e.,

**1**.

**2.**P(S) = 1

**3**. If A and B are mutually exclusive ( Disjoint i.e., )

= P(A)+P(B)**Results of Probability :**. for any two events A & B ,

1

**2**. If denotes compliment of event A then , = 1-P(A)

**3.**For any two events A & B ,

**4**. For any three events A, B & C ,