Sample Space 🚀



## Definition

Definition first:

<Definition>
**Sample Space** 

The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $\Omega$.
</Definition>

So,
- A sample space $\Omega$ is a set;
- Its members are all possible outcomes.


<Definition>
**Sample Space** 

The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $\Omega$.
</Definition>


**Sample Space** 

The set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by $\Omega$.


So,
- A sample space $\Omega$ is a set;
- Its members are all possible outcomes.




A sample space at the "right" granularity, has outcomes which are suit for you study. Also, this sample space should not have irrelevant details.




**Rolling a Dice** 🎲 .  
You want to study the change of getting a number greater than 3, then which one is a more appropriate sample space?  
You are sure that the result of a dice has nothing to do with the weather of tomorrow.

<MCQuest correct="A" totalAttempts="1">
* $\{$ result is greater then 3, result is not greater then 3 $\}$
* $\{$ result is greater then 3 and it rains, result is greater then 3 and it doesn't rain, result is not greater then 3 $\}$
</MCQuest>


**Rolling a Dice** 🎲 .  
You want to study the change of getting a number greater than 3, then which one is a more appropriate sample space?  
You are sure that the result of a dice has nothing to do with the weather of tomorrow.

<MCQuest correct="A" totalAttempts="1">
* $\{$ result is greater then 3, result is not greater then 3 $\}$
* $\{$ result is greater then 3 and it rains, result is greater then 3 and it doesn't rain, result is not greater then 3 $\}$
</MCQuest>


* $\{$ result is greater then 3, result is not greater then 3 $\}$
* $\{$ result is greater then 3 and it rains, result is greater then 3 and it doesn't rain, result is not greater then 3 $\}$




## Exercises
Now let have some exercise about the Sample Space.




For the experiment of flipping a coin, and for each one of the following choices, determine whether we have a legitimate sample space:  (from MITx 6.431x)
<MCQuestList correct="A;B" totalAttempts="1" >
$\Omega=\{$ Heads and it is raining, Heads and it is not raining, Tails $\}$
-----
* Yes
* No
~~~~~

$\Omega=\{$ Heads and it is raining, Tails and it is not raining, Tails $\}$
-----
* Yes
* No
</MCQuestList>


For the experiment of flipping a coin, and for each one of the following choices, determine whether we have a legitimate sample space:  (from MITx 6.431x)

<MCQuestList correct="A;B" totalAttempts="1" >
$\Omega=\{$ Heads and it is raining, Heads and it is not raining, Tails $\}$
-----
* Yes
* No
~~~~~

$\Omega=\{$ Heads and it is raining, Tails and it is not raining, Tails $\}$
-----
* Yes
* No
</MCQuestList>


$\Omega=\{$ Heads and it is raining, Heads and it is not raining, Tails $\}$
-----
* Yes
* No
~~~~~

$\Omega=\{$ Heads and it is raining, Tails and it is not raining, Tails $\}$
-----
* Yes
* No



$\Omega=\{$ Heads and it is raining, Heads and it is not raining, Tails $\}$



$\Omega=\{$ Heads and it is raining, Tails and it is not raining, Tails $\}$



Back to our **Rolling a Dice** 🎲 , which is the sample space of this experiment?
<MCQuest correct="A" totalAttempts="1">
* $\{1,2,3,4,5,6\}$
* $\{1,2,3,4,5\}$
* $\{6\}$
* None of above.
</MCQuest>


Back to our **Rolling a Dice** 🎲 , which is the sample space of this experiment?

<MCQuest correct="A" totalAttempts="1">
* $\{1,2,3,4,5,6\}$
* $\{1,2,3,4,5\}$
* $\{6\}$
* None of above.
</MCQuest>


* $\{1,2,3,4,5,6\}$
* $\{1,2,3,4,5\}$
* $\{6\}$
* None of above.




## Legitimate Sample Space
When we say $\{1,2,3,4,5\}$ is not a sample space of our Rolling a Dice experiment, we mean it's not a legitimate sample space.

But why? Are there any criteria?  
Yes.  

A legitimate sample space has two properties:
- Mutually exclusive,
- collectively exhaustive.




### Mutually Exclusive
What we say a sample space is mutually exclusive, we mean at the end of the experiment, there can only be one of the outcomes that has happened.




**Rolling a Dice** 🎲 .  
Which one of the followings is **not** a mutually exclusive sample space? 

<MCQuest correct="C" totalAttempts="2">
* $\{$ odd numbers,even numbers $\}$
* $\{$ integer $\}$
* $\{1 \text{ or } 2, 3 \text{ or } 4, 5 \text{ or } 6, 6\}$
* $\{$ real numbers $\}$
</MCQuest>


**Rolling a Dice** 🎲 .  
Which one of the followings is **not** a mutually exclusive sample space?

<MCQuest correct="C" totalAttempts="2">
* $\{$ odd numbers,even numbers $\}$
* $\{$ integer $\}$
* $\{1 \text{ or } 2, 3 \text{ or } 4, 5 \text{ or } 6, 6\}$
* $\{$ real numbers $\}$
</MCQuest>


* $\{$ odd numbers,even numbers $\}$
* $\{$ integer $\}$
* $\{1 \text{ or } 2, 3 \text{ or } 4, 5 \text{ or } 6, 6\}$
* $\{$ real numbers $\}$




### Collectively Exhaustive

At the end of the experiment, there is always one of the outcomes that has happened, then we say this sample space is collectively exhaustive.

So the sample space $\{1,2,3,4,5\}$ is not collectively exhaustive for the Rolling a Dice: 
you might roll a 6, but it's not in the sample space.




**Rolling a Dice** 🎲 .  
Which one of the followings is **not** a collectively exhaustive sample space? 

<MCQuest correct="C" totalAttempts="2">
* $\{1,2,3,4,5,6,7\}$
* $\{1,2,3,4,5,6\}$
* $\{1,2,3,4,5\}$
* $\{1 \text{ or } 2, 3 \text{ or } 4, 5 \text{ or } 6 \}$
</MCQuest>


**Rolling a Dice** 🎲 .  
Which one of the followings is **not** a collectively exhaustive sample space?


* $\{1,2,3,4,5,6,7\}$
* $\{1,2,3,4,5,6\}$
* $\{1,2,3,4,5\}$
* $\{1 \text{ or } 2, 3 \text{ or } 4, 5 \text{ or } 6 \}$




## Appropriate Sample Space
You may notice, for an experiment, there might be many legitimate sample spaces.

<Idea>
The sample space of an experiment is decided by you. An experiment may have many legitimate sample spaces.
</Idea>

But not all the sample spaces are appropriate.
A good, appropriate sample space, must be:
- Legitimate
- At the "right" granularity


## Appropriate Sample Space
You may notice, for an experiment, there might be many legitimate sample spaces.

<Idea>
The sample space of an experiment is decided by you. An experiment may have many legitimate sample spaces.
</Idea>


The sample space of an experiment is decided by you. An experiment may have many legitimate sample spaces.


But not all the sample spaces are appropriate.
A good, appropriate sample space, must be:
- Legitimate
- At the "right" granularity



**Rolling a Dice** 🎲 .  
You want to study the chance of getting a 6, then which one is a more appropriate sample space?

<MCQuest correct="B" totalAttempts="2">
* $\{$ odd numbers,even numbers $\}$
* $\{1,2,3,4,5,6\}$
* $\{\{1,2,3\} , \{4,5,6\}\}$
</MCQuest>


**Rolling a Dice** 🎲 .  
You want to study the chance of getting a 6, then which one is a more appropriate sample space?

<MCQuest correct="B" totalAttempts="2">
* $\{$ odd numbers,even numbers $\}$
* $\{1,2,3,4,5,6\}$
* $\{\{1,2,3\} , \{4,5,6\}\}$
</MCQuest>


* $\{$ odd numbers,even numbers $\}$
* $\{1,2,3,4,5,6\}$
* $\{\{1,2,3\} , \{4,5,6\}\}$
