You roll a fair die. if you get a number greater than 4, you win $80. if not, you get to roll again. if you get a number greater than 4 the second time, you win $20. otherwise you win nothing. create a probability model for the amount you win at this game.

Event                          Probability

Win First time                 1/3

Win Second time           2/9

No win                           1- 1/3 -2/9= 4/9

And the expected value for this case would be:

Explanation:

For this case the sampling space when we roll a fair die is :

We define the following events

A= Obtain a number >4 , the sample space for the event A is

And the probability for the event A would be

And the expected value if we got a number greater than 4 is $80 o we can define the amount of money for this event like

If we don't win at the first roll we need to roll a second time the die and for this case we can assume that this event would be B and by the complement rule we know that the probability ofr thi event is :

For this case is we roll the die a second time and we got a number >4 we win $20 and nothing if we got a number lower or equal than 4.

So then the probability of obtain a number >4 in the second chance asuming independence would be:

Then we can define the following model

Event                          Probability

Win First time                 1/3

Win Second time           2/9

No win                           1- 1/3 -2/9= 4/9

And the expected value for this case would be: