LMAD: Go Big or Go Home

On Let's Make A Deal, there is a game played for a car called "Go Big or Go Home." Here is how it is played:
On the game board are various spaces with prizes, plus one GO BIG space (top left), and two GO HOME spaces (bottom right; left side).
Players begin on the GO BIG space, and roll a single 6-sided die to advance clockwise around the board.
Whatever a player lands on, it is added to their prize collection, and the space then becomes a GO HOME.
Landing on a GO HOME loses all prizes and the game is over.
Players attempt to circle the board and land on GO BIG, winning the car, plus their prize collection.
Players may quit at any time if they fear landing on a GO HOME.
If the GO BIG space is passed, players may continue for a second lap around the board, with more GO HOME spaces to contend with.

Would figuring out the probability of winning (GO BIG) use a Markov chain? Or is brute force a better way to go?
Harder question: When should a player stop? Is taking a second lap worth the risk?

what do i need to roll to take that lady home? am i right!?

Quote: sodawater

what do i need to roll to take that lady home? am i right!?

A time machine - this looks like a photo from the original LMAD

Naw that's Tiffany >>? something or other from the current show. (Aw who needs a last name. Or a first name really)