(from Sholander) In a standard clover-leaf interchange, there are four ramps for making right-hand turns, and inside these four ramps, there are four more ramps for making left-hand turns. Your car approaches the interchange from the south. A mechanism has been installed so that at each point where there exists a choice of directions, the car turns to the right with fixed probability [math]r[/math].

• If [math]r = 1/2[/math], what is your chance of emerging from the interchange going west?
• Find the value of [math]r[/math] that maximizes your chance of a westward departure from the interchange.

Notes