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Trial

When an experiment is repeated under similar conditions and it does not give the same result each time but may result in any one of the several possible outcomes, the experiment is called a trial and the outcomes are called cases. The number of times the experiment is repeated is called the number of trials.
 
Example
  1. One toss of a coin is a trial when the coin is tossed 5 times.
  2. One throw of a die is a trial when the die is thrown 4 times.

Different types of events

A subset of the sample space S is called an event.
 
Example

When a die is thrown, sample space S = {1, 2, 3, 4, 5, 6}.

Let A = {1, 3, 5}, here A is the event of occurrence of an odd number.

 

B = {5, 6}, here B is the event of the occurrence of a number greater than 4.
 

Simple event or elementary event

An event is called a simple event if it is a singleton subset of the sample space S.

Mixed event or compound event or composite event

A subset of the sample space S which contains more than one element is called a mixed event.
Example: When a die is thrown, sample space S = {1, 2, 3, 4, 5, 6}.
Let A = {1, 3, 5} = the event of occurrence of an odd number and B = {5, 6} = the event of occurrence of a number greater than 4. Here A and B are mixed events.

Equally likely cases (events)

Cases (outcomes) are said to be equally likely when we have no reason to believe that one is more likely to occur than the other. Thus when an unbiased die is thrown, all the six faces 1, 2, 3, 4, 5, and 6 are equally likely to come up.
 
Similarly, when an unbiased coin is tossed occurrences of head and tail are equally likely cases.

Exhaustive cases (events)

For a random experiment A, set of cases (events) is said to be exhaustive if one of them must necessarily happen every time the experiment is performed.
 
For example, when a die is thrown cases (events) 1, 2, 3, 4, 5, 6, form an exhaustive set of cases (events).

Mutually exclusive or disjoint events

Two or more events are said to be mutually exclusive if one of them occurs, other cannot occur. Thus two or more events are said to be mutually exclusive if no two of them can occur together.
 
Thus events A1A2,..., An are mutually exclusive if and only if Ai  Aj = φ for i ≠ j.

Independent or mutually independent events

Two or more events are said to be independent of occurrence or non-occurrence of any of them does not affect the probability of occurrence or non-occurrence of other events.
 
In other words two or more events are said to be independent if occurrence or non-occurrence of any of them does not influence the occurrence or non-occurrence of other events.
 
Example

When two cards are drawn out of a full pack of 52 playing cards with replacement (the first card drawn is put back in the pack and then the second card is drawn), then the event of occurrence of a king in the first draw and the event of occurrence of a king in the second draw are independent events because the probability of drawing a king in the second draw is 4/52 whether a king is drawn in the first draw or not. But if the two cards are drawn without replacement then the two events are not independent.
 

 

Note: By definition of independent events, it is clear that if A and B are independent events, then
  • A and B′ are independent events.
  • A′ and B are independent events.
  • A′ and B′ are independent events.
  • Non-impossible mutually exclusive events are not independent and non-impossible independent events are not mutually exclusive.




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