Use Bayes Theorem to solve the following:

1. A factory production line is manufacturing bolts using three machines, A, B and C. Of the total output, machine A is responsible for 25%, machine B for 35% and machine C for the rest. It is known from previous experience with the machines that 5% of the output from machine A is defective, 4% from machine B and 2% from machine C. A bolt is chosen at random from the production line and found to be defective. What is the probability that it came from Machine A?

2. Two coins are tossed. Find the conditional probability that two Heads will occur given that at least one occurs.

3. Suppose you are on a game show, and you are given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say Door A [but the door is not opened], and the host, who knows what is behind the doors, opens Door B, which has a goat. He then says to you, "Do you want to pick Door C?" Is it to your advantage to switch your choice?