That's a very good explanation. I tried to generalize the problem to the case of partial additional knowledge about one's intersection, and I invite you to take a look at it to see if it makes the same kind of error. For the case of "ignorance about one's intersection", my solution yields "continue with probability 2/3 at any intersection", just the same as everyone else, and it does so by introducing the parameter r for "probability of guessing intersection correctly". In the problem as stated, r=1/2.