Door 4? No, no, door 7

You find yourself on a game show and there are 10 doors. Behind nine of them are assorted, random farm animals including a variety of pigs, goats, and geese, but behind one is a brand-new, shiny, gold car made of real gold!

The game proceeds as follows:

  • You pick a door.
  • The game show host then opens a door you didn't choose that he knows has only farm animals behind it.
  • You are then given the option to switch to a different unopened door, after which he opens another door with farm animals behind it.

This process continues until there are only two doors left, one being your current choice. (Every time the host opens a door, you are given another option to switch or stick with the one you currently have selected.)

If you play optimally, your chances of winning the car are of the form \( \frac ab\), where \(a\) and \(b\) are coprime positive integers.

Find \(a+b\).

Details and Assumptions:

  • You know ahead of time that this process will continue until you are down to only two doors.

  • Assume that you prefer the car over any of the farm animals!

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