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Events


An event of a random experiment is defined as a subset of the sample space of the random experiment.
Consider the experiment of throwing a dice once. The sample space corresponding to this experiment is S = {1, 2, 3, 4, 5, 6}. Suppose we want the outcomes to be only multiples of 3. We find that 3 and 6 are the only elements of S corresponding to the occurrence of this happening (event). These two elements form the set E = {3, 6}. Thus we see that E is a subset of S. If the outcome of an experiment is an element of an event E, we say that the event E has occurred.

Types of Events:
An event is called an elementary (or simple) event, if it contains only one sample point. Consider the event of getting a number less than 2. We write A = {1}. Thus A is called an elementary event.

An event is said to be a compound event if it has more than one sample point. Consider the event of getting odd numbers as outcomes, then we can write B = {1, 3, 5}. Thus B is a compound event.

An event is called an impossible event, if it can never occur. In the above example, the event C = {7} of getting '7' is an impossible event. On the other hand, an event which is sure to occur is called a sure event. In the above example of rolling a die, the event D of getting a number less than 7 is a sure event. A sure event is also called a certain event.

For example, in the random experiment of throwing two dice:

The sample space is

  1. event of getting sum 4 =
                                        = {(1,3), (2,2), (3,1)}
  2. event of getting sum 8 = 
                                         = {(2,6), (3,5), (4,4), (5,3), (6,2)}
Example
There are 2 children in a family. Find the events that:
  1. both children are boys
  2. only one of the children is a girl
  3. there is at least one girl
  4. the older child is a boy.
Solution
S = {BB, BG, GB, GG}

 

  1. Let A be the event that both children are boys. A = {BB}
    A is a Simple event
  2. Let B be the event that only one of the children is a girl. B = {BG, GB}
  1. Let C be the event that there is at least one girl. C = {BG, GB, GG}

  1. Let D be the event that the older child is a boy. D = {BB, BG}

B, C and D are compound events

Equally Likely Outcomes
The outcomes of a random experiment are called equally likely, if all of them have equal preferences. In the experiment of tossing a unbiased coin, the outcomes, 'Head' and 'Tail' are equally likely.

Exhaustive Outcomes
The outcomes of a random experiment are called exhaustive, if they cover all the possible outcomes of the experiment. In the experiment of rolling a die, the outcomes 1,2,3,4,5,6 are exhaustive.




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