throwing a die. Let E denotes the event “ a number less than appears”. If actually ‘ ’ had appeared on the die then we say that event E has occurred. As a matter of fact if outcomes are or , we say that event E has occurred Thus, the event E of a sample space S is said to have occurred if the outcome ω of the experiment is such that ω ∈ E.
If the outcome ω is such that ω ∉ E, we say that the event E has not occurred. . . Types of events Events can be classified into various types on the basis of the elements they have.
. Impossible and Sure Events The empty set φ and the sample space S describe events. In fact φ is called an impossible event and S, i.e., the whole sample space is called the sure event . To understand these let us consider the experiment of rolling a die.
The associated sample space is S = { , , , , , } Let E be the event “ the number appears on the die is a multiple of ”. Can you write the subset associated with the event E? Clearly no outcome satisfies the condition given in the event, i.e., no element of the sample space ensures the occurrence of the event E. Thus, we say that the empty set only correspond to the event E.
In other words we can say that it is impossible to have a multiple of on the upper face of the die. Thus, the event E = φ is an impossible event. Now let us take up another event F “the number turns up is odd or even”. Clearly PROBABILITY F = { , , , , , ,} = S, i.e., all outcomes of the experiment ensure the occurrence of the event F.
Thus, the event F = S is a sure event. . Simple Event If an event E has only one sample point of a sample space, it is called a simple (or elementary ) event