At Luigi's House of Random Pizza, you can order only one thing: A medium pizza with three random topping layers chosen by Luigi himself. Luigi has 5 different toppings: Pepperoni, Mushroom, Anchovie, Onion, and Pepper. For example, you might get: Pepperoni-Onion-Mushroom, or maybe Mushroom-Pepperoni-Onion (the order of the layers matters to Luigi) or if you are especially lucky then Anchovie-Anchovie-Anchovie.

1. What is the probability that you get a pizza with either two or three distinct toppings? (For example, Mushroom-Onion-Mushroom or Pepper-Onion-Mushroom.)

2. What is the probability that Luigi gives you a pizza with exactly two distinct toppings (occurring in any order)?

3. Suppose (for this problem only) that Luigi is a moody fellow - equally likely to be in a good mood or a bad mood on any given day. When he is in a good mood he picks among his 5 toppings randomly and with equal probability as before. But now, when he is in a bad mood, he gives you an Anchovie-Anchovie-Anchovie pizza. One day, you pick up your pizza and it is, indeed, Anchovie-Anchovie-Anchovie. What is the probability that Luigi was in a bad mood when he made your pizza?