Question

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.

Answer 1

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:

`E(X) = (1)/(3) *80 + (2)/(9) *20 +(4)/(9)*0 = (280)/(9)`

Step-by-step explanation:

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

`S= [1,2,3,4,5,6]`

We define the following events

A= Obtain a number >4 , the sample space for the event A is
`S_A =[5,6]`

And the probability for the event A would be
`p_A = (2)/(6) =(1)/(3)`

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
`X_A = 80`

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 :

`p_B = 1-p_A = 1-(1)/(3)= (2)/(3)`

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:

`P_(SB) = (2)/(3) *(1)/(3)= (2)/(9)`

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:

`E(X) = (1)/(3) *80 + (2)/(9) *20 +(4)/(9)*0 = (280)/(9)`

Answer 2

The total no. of possible outcomes of rolling a fair die = 6

The possible outcomes are 1, 2, 3, 4, 5 and 6

The event to occur is getting a no. greater than 4.

The no. of possible outcomes to get a no. greater than 4 = 2

So, probability of getting a no. greater than 4 = 2/6 = 1/3

If you get a no. greater than 4 for the first time, you win $80.

So, the probability to win $80 is 1/3.

The probability to roll the die again = 1 - (1/3) = 2/3

In the second chance, if you get a no. greater than 4 you win $20, else nothing.

Since the two events are dependent, P(X and Y) = P(X) * P(Y)

Probability of getting a no. greater than 4 in the second chance = (2/3) * (1/3) =2/9

Probability of win (either in 1st or 2nd chance) = 1/3 + 2/9 =5/9

Please check attachment for probability model.