Reduced Sample Space

Reduced sample space is used during solving probability problems. When additional information is mentioned about the experiment, then the sample space “S” will be reduced sample space. The symbol for reduced sample space is “Sr”. To understand what additional information means, consider a die has been rolled and it has been informed that the experiment has produced only odd face.  This mentioned information is known as additional information. Therefore, it is now clear that the sample space reduces to reduced sample space only because of additional information. So, whenever there is information been given than go for the reduced sample space instead of complete sample space.

Examples of Reduced Sample Space:

Example 1:

A die has been rolled where additional information is given that the experiment has produced an even face.


Consider the first part of the statement i.e., “A die has been rolled”. When a die is rolled, there are six possible outcomes.

So, complete sample space is comprise of six events

S = {1, 2, 3, 4, 5, 6}

Now consider the second part of the statement i.e., the additional information “even face has occurred”.

Now a complete sample space reduces to reduce sample space.

Sr = {2, 4, 6}

In this example, information is given that the even face has occurred. It may be 2, 4 or 6. Specific face is not mentioned in this case. If information is given that certain specific even face has occurred then it is no more a situation of probability. Because, when it given that even face occurred than there is still something that is hidden from the person. This hidden detail makes actual outcome not known to the observer.

Example 2:

A fair die is rolled. Find the reduced sample space when it is given that the appearing face is less than 4.


Sample space consists of six events when a die is rolled.

Sample space = S = {1, 2, 3, 4, 5, 6}

Additional information is given that the occurring face is less than 4. Therefore, sample space changes to reduced sample space.

Sr = {1, 2, 3}




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