## Classical definition of probability.

Introduction :
The chances of happening or non happening of an event when expressed quantitatively is called probability. There  are mainly two definitions of probability.
(i) Experimental                                        (ii) Theoretical
In this chapter, we will study about these two types of probability. Experimental probability approach is when we toss the coin one – million times or 10 million times, we observe that as the number of tosses increases the experimental probability of a head or a tail comes close to 0.5. This is experimental approach to probability. The basic difference between the experimental and theoretical approach to probability is that former is based on what has actually happened but later is based on prediction of what will happen

Probability – A Theoretical Approach

If an event A occurs m times and does not occurs n times. Then, the probability of occurence of event A, denoted by P(A) is given by

Sum of the Probabilities of All the Elementary Event of an Experiment
If E1 , E2 ..........., En be the n elementary events associated with a random experiment having exactly n outcomes, then P(E1) + P(E2) + .......... + P(En) = 1.
For example, when we toss a coin, the resulting elementary events are H and T.

## Simple problems on single events (not using set notation).

Illustration : 1
A box contains 3 blue, 2 white and 4 red marbles. If a marble is drawn at random from the box, what is the probability that it will be
(i) white ?                  (ii) blue ?            (iii) Red ?

Solution :
Saying that a marble is drawn at random is a short way of saying that all the marbles are equally likely to be drawn.
Therefore, the number of possible outcomes = 3 + 2 + 4 = 9
Let W denote the event ‘the marble is white’ , B denotes the event ‘the marble is blue’ and R denote the event ‘marble is red’.

Illustration : 2
Two coins are tossed simultaneously. Find the probability of getting
Solution :
Let H denotes head and T denotes tail.
On tossing two coins simultaneously, all possible outcomes HH, HT, TH, TT = 4.

Illustration : 3
17 cards numbered 1, 2, 3, ........, 17 are put in a box and mixed throughly. One person draws a card from the box. Find the probability that the number on the card is
(i) odd        (ii) a prime    (iii) divisible by 3    (iv) divisible by 3 and 2 both.
Solution :
(i) There are 9 odd numbered cards, namely 1, 3, 5, 7, 9, 11, 13, 15, 17. Out of these 9 cards one card can be drawn in 9 ways.
Favourable number of elementary events = 9
Total number of elementary events = 17.

(ii)     There are 7 prime numbered cards, namely. 2, 3, 5, 7, 11, 13, 17. Out of these 7 cards one card can be chosen in 7 ways.
Favourable number of elementary events = 7
Total number of elementary events = 17

.

(iv)     If a number is divisible by both 3 and 2, then it is a multiple of 6. In cards bearing  number 1, 2, 3, ........17 there are only 2 cards which bear a number divisible by 3 and 2  both i.e., by 6.  These cards bear numbers 6 and 12.
Favourable number of elementary events = 2