This is the first of three posts. The second has "the answer".
Thank you all for responding. As a group you have a great sense of humor. I laughed out loud many times while reading. I particularly liked Flynbys idea of staying in the tunnel.
Clearly a case of thinking outside the box.
I must admit that when I first encountered this problem I got the answer wrong. Not only did I get the answer wrong but I was completely certain that I had it correct. (As some of you may have pieced together from some of my previous SR posts, this is not an altogether uncommon event.)
Several folks got the correct answer. This was far more than I had predicted. My suspicion is that some of them had help via Google, but that's unimportant really. The important thing is to examine your own thinking. I think the wisdom from the poem above "decide for yourself" points in that direction.
I believe that more than 95% of the population who first encounter this problem fail to get the correct answer. My guess is that most of those people believe (as I did) that the chances for A go from 33% to 50% and that it does not matter if A stays or switches cells with B. That's just my estimate, but I bet I'm fairly close in that estimate.
You might be interested to know that many professional mathematicians have gotten the answer wrong while at the very same time ridiculing those with the correct answer. Also, getting the correct answer does not seem to be related to intelligence. BTW, A famous mathematician named Paul Erdos got it wrong initially! Took him a while to find a way to understand it.
Next, the "answer".