Are both sample spaces the same or do they mean different things?

[vcsharp2003]

Homework Statement

A coin is tossed two times. What will be the sample space for this experiment?

Relevant Equations

None

I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up.

$$ S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \} $$

$$ S = \{ (i,j) : i \in \{H,T\}, j \in \{H,T\} \} $$

 

[pasmith]

Homework Statement:: A coin is tossed two times. What will be the sample space for this experiment?
Relevant Equations:: None

I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up.

$$ S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \} $$

$$ S = \{ (i,j) : i \in \{H,T\}, j \in \{H,T\} \} $$

The order in which members of a set are listed is irrelevant, and multiple occurences of the same element in the list are redundant, so the first is [tex]\{\{H\}, \{H,T\}, \{T\} \}.[/tex] Can this be right? If the two tosses have different outcomes, should we be able to distinguish between a head followed by a tail as opposed to a tail followed by a head?

 

[vcsharp2003]

The order in which members of a set are listed is irrelevant, and multiple occurences of the same element in the list are redundant, so the first is [tex]\{\{H\}, \{H,T\}, \{T\} \}.[/tex] Can this be right? If the two tosses have different outcomes, should we be able to distinguish between a head followed by a tail as opposed to a tail followed by a head?

Ok, I get it.

If the first form is used then we will get one of the members of set S as ##\{ H,H \}## which is the same as ##\{ H\}##. Therefore, the first form is not correct and the second form is correct.

It seems to me that we're essentially trying to get Cartesian Product as the set S i.e. ##\{ H,T\} \times \{ H,T\} ## for which we always use the second form. Also we know that in Cartesian Product the order is important, which would account for order of H and T in a pair of values.

 

[Mark44]

Homework Statement:: A coin is tossed two times. What will be the sample space for this experiment?
Relevant Equations:: None

I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up.

$$ S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \} $$

$$ S = \{ (i,j) : i \in \{H,T\}, j \in \{H,T\} \} $$

Your "two different forms" are actually identical.
Edit: I see now that the first version has ordered pairs (in parentheses) and the second has sets (in braces).

The order in which members of a set are listed is irrelevant, and multiple occurences of the same element in the list are redundant, so the first is [tex]\{\{H\}, \{H,T\}, \{T\} \}.[/tex]

@pasmith, I'm not what you're trying to say here, but the sample space should be a set of events, not a set of sets. IOW, looking like this: ##\{ (H, H), (H, T), (T, H), (T, T) \}##. Each of the listed pairs (events) is equally likely.

 

[pasmith]

Your "two different forms" are actually identical.

The first form uses braces [itex]\{i,j\}[/itex]. That indicates a set, not an ordered pair. The second form is correct.

 

[Mark44]

The first form uses braces {i,j}.

A detail that my old eyes missed.