The word ‘probability’ in statistics means the ‘chance’ or ‘possibility’ or ‘likelihood’. Probability is a numerical (statistical) measure of the chance or possibility or likelihood that a particular event will occur or not.
It always lies between the value zero and one. If there is no any possibility that an event will occur then the probability of event is zero but if there is absolute possibility of occurrence of an event then its probability is equal to one. Initially, it was used in the game of chance but nowadays, it is used in most of all disciplines like; statistics, economics, business, engineering, industry etc. It is commonly used in decision making.
Terminology in Probability in statistics
The Commonly used terms in probability in statistics are as follows
- Trial and event (case).
- Sample space.
- Exhaustive cases.
- Favourable event.
- Equally likely event.
- Mutually exclusive event.
- Independent event.
- Dependent event.
- Sure event.
- Impossible event.
- Simple event.
- Compound/composite event.
- Permutation.
- Combination.
- Random experiment.
Random experiments
Any operation that results in two or more outcomes is called an experiment and the experiment of which result is not unique and has one of the several possible outcomes under the similar condition is known as random experiment. Tossing of a fair coin, rolling of a die, drawing cards from well shuffled pack of playing cards etc are some examples of random experiment.
What is Trial and Events in probability in statistics
- Performing a random experiment is a trial. On the other hand, the results or outcomes of an experiment are called an events.
- For example; tossing of an unbiased coin is a trial and getting either head or tail is an event.
- Sample space:
- The set of all possible outcomes of an experiment is called sample space. Thus, an event is a subset of sample space.
- In tossing an unbiased coin once, the outcomes head and tail is the sample space.
Exhaustive Cases (events):
- Cases are said to be exhaustive when it includes all possible outcomes of a random experiment.
- In tossing an unbiased coin once, the exhaustive cases are Head and Tail. Therefore the total exhaustive number of cases is 2.
- Favorable Events (cases):
- The number of outcomes of a random experiment which results in the happening of an event is called favorable events or cases.
- For example; in drawing a card from a pack of 52playing cards, the no. of favorable cases of drawing Ace is 4.
Equally Likely Events in probability
- The events are said to be equally likely, if the probability of occurrence of any event is equal to the others.
- In other words, events are said to be equally likely, if anyone of them can’t be expected to occur in preference to others.
- For example; in coin toss experiment, the chance of getting head or tail is equal thus they are equally likely events.
- Mutually Exclusive Events:
- If two or more than two events can’t occur simultaneously at the same time in the same trial then the events are said to be mutually exclusive events.
- For example; in coin toss experiment; if one event head occurs then the other event tail can’t occur thus these events are called mutually exclusive events.
Simple and compound events in probability
- Any single event that results in a random experiment is called simple event. It is also called elementary event.
- On the other hand, the union of the elementary events is called compound or composite event.
- For example; if two coins are tossed then occurrence of two heads only (HH) is called simple event but the union of the elementary events like union of (HH,HT) or union of (HH,HT,TH) etc.
Dependent and independent Events in statistics
- Events are said to be dependent, if the occurrence of one event affects the occurrence of other events.
- For example, if a card is playing without replacement, the occurrence of a king affects the occurrence of other cards.
- On the other hand, Events are said to be independent if the occurrence of one event doesn’t affect the occurrence of other events.
- For example, if a pack of card is playing with replacement, then the occurrence of Ace does not affect the occurrence of other cards.
Sure Events in probability
- An event which occurs absolutely (surely) is called sure event. The probability of occurrence of sure event is always one.
- For example, in throwing a die, an event of getting a number less than 7 is a sure event.
- Impossible event:
- An event which is not possible to occur in an experiment is called impossible event.
- For example: in throwing a fair die, an event of getting the number 7 is impossible event.
Fundamental Principle of Counting:
Permutation:
The process of arranging different things by taking all of them at a time is called permutation. The number of permutation of n things, taken ‘r’ at a time, is given by,
Similarly , the number of permutation of ‘n’ different things taken ‘r’ at a time when each things may repeat ‘p’ times, ‘q’ times, ‘r’ times and so on is given by,
If the ‘n’ things are arranged in a circular way then the permutation is given by
P = (n-1)!
If things are arranged in a straight line then the permutation is given by
P = n!
Combination:
The process of selecting the objects out of group of objects without any order of arrangement is called combination. A combination of ‘n’ different objects taken ‘r’ objects at a time without considering the order of arrangement is given by:
Definition (Approach) of Probability:
The common approaches of probability are;
- Mathematical or Classical or Priori Approach.
- Statistical or Empirical or Relative frequency Approach.
- Subjective Approach.