Before discussing the theory of probability, let us have an understanding of the following terms :
9.2.1. Random Experiment or Trial : If an experiment or trial can be repeated under the same conditions, any number of times and it is possible to count the total number of outcomes, but individual result i.e. individual outcome is not predictable.
Suppose we toss a coin. It is not possible to predict exactly the outcomes. The outcome may be either head up or tail up. Thus an action or an operation which can produce any result or outcome is called a random experiment or a trial.
9.2.2. Event : Any possible outcome of a random experiment is called an event. Performing an experiment is called trial and outcomes are termed as events.
An event whose occurrence is inevitable when a certain random experiment is performed, is called a sure event or certain event. At the same time, an event which can never occur when a certain random experiment is performed is called an impossible event. The events may be simple or composite. An event is called simple if it corresponds to a single possible outcome. For example, in rolling a die, the chance of getting 2 is a simple event. Further in tossing a die, chance of getting event numbers (1, 3, 5) are compound event.
9.2.3. Sample space The set or aggregate of all possible outcomes is known as sample space. For example, when we roll a die, the possible outcomes are 1, 2, 3, 4, 5, and 6 ; one and only one face come upwards. Thus, all the outcomes— 1, 2, 3, 4, 5 and 6 are sample space. And each possible outcome or element in a sample space called sample point.
9.2.4. Mutually exclusive events or cases : Two events are said to be mutually exclusive if the occurrence of one of them excludes the possibility of the occurrence of the other in a single observation. The occurrence of one event prevents the occurrence of the other event. As such, mutually exclusive events are those events, the occurrence of which prevents the possibility of the other to occur. All simple events are mutually exclusive. Thus, if a coin is tossed, either the head can be up or tail can be up; but both cannot be up at the same time.
Similarly, in one throw of a die, an even and odd number cannot come up at the same time. Thus two or more events are considered mutually exclusive if the events cannot occur together.
9.2.5. Equally likely events : The outcomes are said to be equally likely when one does not occur more often than the others.
That is, two or more events are said to be equally likely if the chance of their happening is equal. Thus, in a throw of a die the coming up of 1, 2, 3, 4, 5 and 6 is equally likely. For example, head and tail are equally likely events in tossing an unbiased coin.
9.2.6. Exhaustive events The total number of possible outcomes of a random experiment is called exhaustive events. The group of events is exhaustive, as there is no other possible outcome. Thus tossing a coin, the possible outcome are head or tail ; exhaustive events are two. Similarly throwing a die, the outcomes are 1, 2, 3, 4, 5 and 6. In case of two coins, the possible number of outcomes are 4 i.e. (22), i.e., HH, HT TH and TT. In case of 3 coins, the possible outcomes are 23=8 and so on. Thus, in a throw of n” coin, the exhaustive number of case is 2n.
9.2.7. Independent Events A set of events is said to be independent, if the occurrence of any one of them does not, in any way, affect the Occurrence of any other in the set. For instance, when we toss a coin twice, the result of the second toss will in no way be affected by the result of the first toss.
9.2.8. Dependent Events Two events are said to be dependent, if the occurrence or non-occurrence of one event in any trial affects the probability of the other subsequent trials. If the occurrence of one event affects the happening of the other events, then they are said to be dependent events. For example, the probability of drawing a king from a pack of 52 cards is 4/52, ; the card is not put back ; then the probability of drawing a king again is 3/51. Thus the outcome of the first event affects the outcome of the second event and they are dependent.
But if the card is put back, then the probability of drawing a king is 4/52 and is an independent event.
9.2.9. Simple and Compound Events When a single event take place, the probability of its happening or not happening is known as simple event.
When two or more events take place simultaneously, their occurrence is known as compound event (compound probability) ; for instance, throwing a die.
9.2.10. Complementary Events : The complement of an events, means non-occurrence of A and is denoted by contains those points of the sample space which do not belong to A. For instance let there be two events A and B. A is called the complementary event of B and vice verse, if A and B are mutually exclusive and exhaustive.
9.2.11. Favourable Cases
The number of outcomes which result in the happening of a desired event are called favourable cases to the event. For example, in drawing a card from a pack of cards, the cases favourable to “getting a diamond” are 13 and to “getting an ace of spade” is only one. Take another example, in a single throw of a dice the number of favourable cases of getting an odd number are three -1,3 and 5.