From a comment by aws17576 on MetaFilter:
By the way, I wholeheartedly endorse Persi Diaconis's comment that probability is one area where even experts can easily be fooled. This was demonstrated to me in grad school when my advisor, addressing a roomful of mathematicians, posed this problem:
Person A flips a coin repeatedly, stopping the first time two heads in a row appear. Person B flips a coin repeatedly, stopping the first time a head and then a tail appear in a row. Who will flip the coin more times on average -- A, B, or is there no difference?
He let everyone think for a moment, then took a show of hands. Almost everyone got it wrong.