One easy counter-example is "50% [well, technically a bit less, but ignore that for now] of humans are capable of getting pregnant. You are human. Therefore, there is a 50% chance that you are capable of getting pregnant." This ignores the fact that there is another very obvious discriminant - namely, sex - between people who are capable of getting pregnant and people who are not. If you are male, your chance is 0%, if you are female your chance is [close to] 100%, and it is fairly easy to tell without actually trying to get pregnant which one you are.

This also comes up all the time in data-science and A/B testing of products. Frequently you'll end up with results like "This change decreased revenue by 3%." Sounds bad, right? Except then you drill into your results and find out that there was a bug in the implementation of the change in IE6, which decreased revenue by 100% on that one browser which happens to be 5% of traffic, and find that your change actually increased revenue by a percent or two, but that one slice cost you all the benefit and more. You fix the bug and happily start making more money.

Always slice your population. Many times blanket probabilities tell you nothing unless you know what's causing them.

Right. An analogy would be saying "people who eat over 6000 calories a day are, as a rule, overweight". It's a logical fallacy to turn around and say "Michael eats over 6000 calories a day, therefore he is overweight". It's statistically very likely, but in this case Michael is a olympic swimmer with 10 gold medals. It's been a long time since my logic class in college, but the rule of 'statistics are great until you try to apply them to an individual' has always stuck in my mind. Probably because statistics are often one step removed from 'stereotypes'.