Sequential Models 🚥



## Sequential Experiment
Many experiments have an inherently sequential character; for example, tossing a coin two times.

In this section we will learn how to describe a sequential model.  
Note:
**the order matters when not specifically stated.**





**Tossing a coin 🪙 two times.**  
Are the following sample spaces appropriate for this experiment? (H for Heads and T for Tails)
<MCQuestList correct="A;B;A" totalAttempts="2" >
1) $\{HH,HT,TH,TT\}$
-----
* Yes
* No
~~~~~

2) $\{HH,TT,HT\}$
-----
* Yes
* No
~~~~~

3) $\{(H,H), (T,T),(H,T),(T,H)\}$
-----
* Yes
* No
</MCQuestList>


**Tossing a coin 🪙 two times.**  
Are the following sample spaces appropriate for this experiment? (H for Heads and T for Tails)


1) $\{HH,HT,TH,TT\}$
-----
* Yes
* No
~~~~~

2) $\{HH,TT,HT\}$
-----
* Yes
* No
~~~~~

3) $\{(H,H), (T,T),(H,T),(T,H)\}$
-----
* Yes
* No




<Idea>
We often use lists to describe the outcomes of a sequential model, like $(x,y,z)$. 
</Idea>



We often use lists to describe the outcomes of a sequential model, like $(x,y,z)$. 




## Tree-Based Sequential Description
Since a sequential experiment is consist of multiple experiment, it is then often useful to describe the experiment and the associated sample space by means of **a tree-based sequential description**.

Here is the tree for tossing a coin two times:
<Mermaid width="300" height="300">
graph LR
    Root --- T
    Root --- H
    T --- T,T
    T --- T,H
    H --- H,T
    H --- H,H
</Mermaid>


## Tree-Based Sequential Description
Since a sequential experiment is consist of multiple experiment, it is then often useful to describe the experiment and the associated sample space by means of **a tree-based sequential description**.

Here is the tree for tossing a coin two times:

<Mermaid width="300" height="300">
graph LR
    Root --- T
    Root --- H
    T --- T,T
    T --- T,H
    H --- H,T
    H --- H,H
</Mermaid>


graph LR
    Root --- T
    Root --- H
    T --- T,T
    T --- T,H
    H --- H,T
    H --- H,H




(This question is from *Probability* by MITx.)  
Paul checks the weather forecast. If the forecast is good, Paul will go out for a walk. If the forecast is bad, then Paul will either stay home or go out. If he goes out, he might either remember or forget his umbrella. In the tree diagram below, identify the leaf that corresponds to the event that the forecast is bad and Paul stays home.

![Question](/probability/01-03-a.jpeg)

<MCQuest totalAttempts="2" correct="B">
* 1
* 2
* 3
* 4
</MCQuest>


(This question is from *Probability* by MITx.)  
Paul checks the weather forecast. If the forecast is good, Paul will go out for a walk. If the forecast is bad, then Paul will either stay home or go out. If he goes out, he might either remember or forget his umbrella. In the tree diagram below, identify the leaf that corresponds to the event that the forecast is bad and Paul stays home.

![Question](/probability/01-03-a.jpeg)