I have a dim view of the human mind's capacity for rational thought. Even so, I was struck by this old experiment, flagged by Robert Frank:

An intriguing example of transparently irrelevant information that affects behavior comes from a 1974 report on an experiment by the psychologists Daniel Kahneman and Amos Tversky. In the experiment, subjects first spun a wheel that supposedly would stop at random on any number between 1 and 100. Then they were asked what percentage of African countries belongs to the United Nations. For one group of subjects, the wheel was rigged to stop on 10; for a second group, on 65. On average, the first group guessed that 25 percent belong to the United Nations, but the second group guessed 45 percent.