1) A good demonstration of how "random" needs to be defined in order to work out probability of certain events:
What is the chance that a random line drawn across a circle will be longer than the radius?
The question totally depends how you draw your line. Possibilities which you can describe in a real-world way for a circle drawn on the ground might include:
Draw two random points inside the circle and draw a line through those. (see where two flies / stones fall...)
Choose a fixed point on the circumference, then a random one elsewhere in the circle and join those. (in effect this is laying a stick across the circle at a variable angle through a given point and a random one eg where stone falls)
Draw a diameter. Randomly choose a point along it and draw a perpendicular through that. (roll a stick along in a straight line so it rests across the circle)
It is relatively easy to show someone who can do some geometry (but not necessarily stats) the answer to the question can vary quite widely (from about 2/3 to about 0.866 or so)
2) A reverse-engineered coin-toss: toss it (say) ten times and write down the result. Work out the probabilty of this exact sequence (1/2^10). A tiny chance, but you just saw it happen with your own eyes!... Every sequence might come up, including ten heads in a row, but it is hard for lay people to get their head round it. As an encore, try to convince them they have just as good a chance of winning the lottery with the numbers 1 through 6 as any other combination.
3) Explaining why medical diagnosis may seem really flawed. A test for disease foo which is 99.9% accurate at identifying those who have it but .1% false-positively diagnoses those who don't really have it may seem to be wrong really so often when the prevalence of the disease is really low (eg 1 in 1000) but many patients are tested for it.
This is one that is best explained with real numbers - imagine 1 million people are tested, so 1000 have the disease, 999 are correctly identified, but 0.1% of 999,000 is 999 who are told they have it but don't. So half those who are told they have it actually do not, despite the high level of accuracy (99.9%) and low level of false posisitive (0.1%). A second (ideally different) test will then separate these groups out.
[incidentally, I chose the numbers because they are easy to work with, of course they do not have to add up to 100% as the accuracy / fales positive rates are independent factors in the test]