I'd go the opposite way, I'd play out an intricate strategy based on the first few boxes, then use probability and high level maths to work out where I'd choose the next box.

....or that's what I'd tell 'em. I'd have a box in my head based on a completely random fact like, how many audience members are wearing red sweaters, or something else utterly stupid, and I wouldn't stop until I'd got a good offer or got to the end - I'd just keep feeding them absolute bollox until the second to last box and when they film the whole pause to open the last box, I'd tell them I made the whole theory up and open it so they couldn't cut it from the show.