Theory of Probability

Random Experiment : - An experiment which repeated under essentially identical conditions, doesn't gives unique results but may be occurred any one of the several possible outcomes is called random experiment. It is also called Trail.

Outcome : - The result of random experiment is called outcome.

Exhaustive cases : - The total number of all possible outcomes of any experiment is called exhaustive cases. It is also called sample space. It is denoted by S

Event : - Single outcome or collection outcomes are called event. In other words, sub set of sample space is called Event. Event is generally denoted by capital letters. An event having no sample point , it is called null event. An event consisting of only one sample point (outcome) of a sample space is called a simple event.

Mutually exclusive events : - If in an experiment the occurrence of an event precludes the happening of all other events then the events are called Mutually exclusive events.

Equally likely Events : - Events are to be said to be equally likely if there is no reason to expect any one in performance to other.


Probability definitions :

I) classical definition : - In a random experiment, n is exhaustive, mutually exclusive and equally likely and m of them favorable to the happening of an event E then the probability of happening of the event E is P(E) = favorable cases / Exhaustive cases (i.e.) P(E) = m / n
The probability of not happening of the event E is P(Ec) = (n - m) / n = 1 - m/n = 1 - P(E).
It is also called Priory or Mathematical definition. It was introduced by Laplace.

Limitations:-
i) Classical definition fails when n is infinite
ii) If fails when events are not equally likely.

II) Empirical definition : - It is also called Relative frequency or Statistical definition. It was introduced by Von Misses.

The table gives the probability of rolling a specific number;
This table gives the probability of rolling a particular number on a set of so many 6-sided dice.
Roll: 1 die - 2 dice - 3 dice - 4 dice
1 -- 1/6 ---- 0 ---- 0 ---- 0
2 -- 1/6 -- 1/36 -- 0 -- 0
3 -- 1/6 -- 2/36 -- 1/216 -- 0
4 -- 1/6 -- 3/36 -- 3/216 -- 1/1296
5 -- 1/6 -- 4/36 -- 6/216 -- 4/1296
6 -- 1/6 -- 5/36 -- 10/216 -- 10/1296
7 --- 0 -- 6/36 -- 15/216 -- 20/1296
8 --- 0 -- 5/36 -- 21/216 -- 35/1296
9 --- 0 -- 4/36 -- 25/216 -- 56/1296
10 -- 0 -- 3/36 -- 27/216 -- 80/1296
11 -- 0 -- 2/36 -- 27/216 -- 104/1296
12 -- 0 -- 1/36 -- 25/216 -- 125/1296
13 -- 0 -- 0 -- 21/216 -- 140/1296
14 -- 0 -- 0 -- 15/216 -- 146/1296
15 -- 0 -- 0 -- 10/216 -- 140/1296
16 -- 0 -- 0 -- 6/216 -- 125/1296
17 -- 0 -- 0 -- 3/216 -- 104/1296
18 -- 0 -- 0 -- 1/216 -- 80/1296
19 -- 0 -- 0 -- 0 -- 56/1296
20 -- 0 -- 0 -- 0 -- 35/1296
21 -- 0 -- 0 -- 0 -- 20/1296
22 -- 0 -- 0 -- 0 -- 10/1296
23 -- 0 -- 0 -- 0 -- 4/1296
24 -- 0 -- 0 -- 0 -- 1/1296