Food for thought

1. Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

2.
A random two-child family whose older child is a boy is chosen. What is the probability that the younger child is a girl?
A random two-child family with at least one boy is chosen. What is the probability that it has a girl?

3. There are three prisoners, A, B, and C, two of whom will be released and the other executed. A asks the warden to tell him the name of one of the others who will be released. As the question is not directly about A's fate, the warden obliges. Assuming the warden's truthfulness, there are now only two possibilities for who will be executed: A, and either B or C. Did A gain any information as to his own fate, that is, does he change his estimate of the chances he will be executed? If the warden says "B will be released" and A could switch fates with C, should he?

4.
Suppose you have three cards:
a black card that is black on both sides,
a white card that is white on both sides, and
a mixed card that is black on one side and white on the other.
You put all of the cards in a hat, pull one out at random, and place it on a table. The side facing up is black. What are the odds that the other side is also black?

And if you haven't already guessed, the answers are not as straightforward as you would think

These paradoxical probability questions are much more interesting than Economics...but they do hurt my brain more X_X