Roll of the dice simulation

Question

I have 3 dice and a $1 wager. I roll the first dice. Whatever the result, I want to roll the second dice such that it is lower than the first. So, if I roll a 3 on the first dice, I want to roll a 1 or a 2 on the second dice. If I'm successful, I get a 50% return on my money. So if I wagered $1, I get $1.50 back. I can stop there or continue with the third dice which I also want to roll lower than the first. If I'm successful, I double my wager, so my $1 becomes $2. If I'm unsuccessful on either the second or third roll, I lose my $1 wager. What are the mathematically expressions for combinations (ie "choose") that show all probabilities for this simulation.

Answer 1

Please clarify:

There is some ambiguity about the third roll. In some cases, the expectation is higher if one rolls a third time (for example, if the first roll is 6) than if one stops after the second roll, and other times the expectation is lower if one rolls a third time (for example, if the first roll is 2) than if one stops after the second roll.

One way to fix this is to ask "What is the expected outcome of the game if one makes a third roll exactly in those situations where the expectation is higher for the outcome after 3 rolls and the expectation for stopping at 2 rolls?"

Let us know if this is really the problem you would like us to solve, and, if so, one of us will solve it, using a tree diagram.