"Consider a regular six-sided die with four green faces and two red faces. The die will be rolled 20 times and the sequence of greens (G) and reds (R) will be recorded. You are asked to select one sequence, from a set of three, and you will win $25 if the sequence you chose appears on successive rolls of the die. Please check the sequence of greens and reds on which you prefer to bet.
1. RGRRR
2. GRGRRR
3. GRRRRR"
65% of the subjects chose 2, even though 1 provides the greatest chance of winning money. (From "Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment," by Tversky and Kahneman.)
Maybe all those subjects had the unconventional goal of avoiding free money. But it seems more likely that they weren't thinking well.