Modified Monty Hall
A mathematician is on a gameshow, and the host gives him a choice of three doors; behind one is a Ferrari, but the other two lead to empty rooms. If he chooses the correct door, the host will open an empty door and give him the chance to choose again. However, if he chooses an incorrect door, the host will open the other empty door and give him the opportunity to choose again with probability \(p\) (otherwise, he will tell him that he has lost).
The mathematician picks a door and the host opens another and gives him a chance to switch. The mathematician, who always makes true statements and is aware of the host's strategy, tells the host that changing does not improve or decrease his probability of winning the Ferrari. What is \(p\)?