I have also had a lecture on this excellent topic.
Unfortunately, my lecture notes are in Czech, but I can translate some paradoxes from there:
Monty Hall is in my opinion the most famous probabilistic paradox. It is described here and on the Wiki well. So I just provide you a way how to make it feel intuitive, to persuade other people that the real probability is computed correctly. It is quite impressive, almost like a magician trick :-)
Take a pack of cards and let someone draw a random card. Tell him that he want to draw the ace of hearts and he should not look on the chosen card. Then show to audience all remaining cards but one which are not the ace of hearts. So then there are two hidden cards: One in your hand and one in his hand. Finally, he may change his first guess. Most people do it and they are most likely correct :-).
There are two players, Alice and Bob, playing a tennis-like game. Every time, one player serves the ball and winning probabilities of the ball depends on the player. Player who first reaches say 11 points wins the match.
Alice serves the first ball. Then there are three possible schemes:
- The winner of the previous ball serves the next.
- The loser of the previous ball serves the next.
- Service is regularly alternating.
One would expect that scheme 1. helps stronger players and scheme 2 helps weaker players. The paradox is that winning probabilities of the match do not depend on the chosen scheme.
Proof sketch: Pregenerate 11 cards with the winner of Alice services (pack A) and 10 cards with the winner of Bob's service (pack B). Then each Alice's (or Bob's) service can be modeled just by drawing a card from the pack A (or B). It can be shown that these 21 cards suffice for any of these 3 presented schemes. And the winner is determined by the cards: there is exactly one player written on at least 11 cards.
I have a bag of candies, there are 123 caramel candies and 321 mint candies. Every morning I randomly draw candies from the pack and eat them while they are all the same. When I take a different kind of candy, I return it back. What is the probability that the last eaten candy will be the caramel one?
Answer: 1/2. (one would expect that it is less than 1/2 since there are less caramel candies)
Proof: It suffices to show that every morning the probability that all caramel candies will be eaten is the same as the probability that all mint candies will be eaten. We can imagine that candies are randomly ordered every morning and I am drawing them from left to right. I eat all caramel candies if the order is "First caramel candies, then mint ones.". I eat all mint candies if the order is the opposite.
Wolf on a circle
There is a wolf at one vertex of a regular n-gon. There is a sheep at every remaining vertex. Each step, the wolf moves to a randomly chosen adjacent vertex and if there is a sheep, the wolf eat it. The wolf ends when it eats n-2 sheep (so there remains just one sheep).
Intuitively, the sheep at the opposite vertex from the wolf is in the best position. The paradox is that all sheep have the same probability of survival.
Proof: Take one sheep S. The wolf will definitely get to an adjacent vertex to S for the first time. This time the sheep on the other adjacent vertex is still not eaten. So for S survival, the wolf have to go around the whole circle without eating S. The probability of going around the whole circle from one vertex adjacent to S to the other without visiting S does not depend on the position of S.
There is a research on two medical cures A, B.
200 people tried the cure A, it helped to 110 people (50 men, 60 women) and did not helped to 90 people (60 men, 30 women).
210 people tried the cure B, it helped to 120 people (30 men and 90 women) and did not helped to 90 people (40 men, 50 women).
So in general, the cure B is better since 120:90 > 110:90.
But if you are a man, you can consider just men statistics: 50:60 > 30:40, so the cure A is more appropriate.
And if you are a woman, you can consider just women statistics: 60:30 > 90:50, so the cure A is again more appropriate.
Shocking, isn't it? :-)