At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semi- circular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is selected uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping. TI T3 T3 TI T2 T3 T2 T2 T1 = Topping I T2 = Topping 2 T3 = Topping 3 (a) For a 2-topping pizza, determine the probability that at least of the pizza is covered by both toppings. (b) For a 3-topping pizza, determine the probability that some region of the pizza with non-zero area is covered by all 3 toppings. (The diagram above shows an example where no region is covered by all 3 toppings.) (c) Suppose that N is a positive integer. For an N-topping pizza, determine the probability, in terms of N, that some region of the pizza with non-zero area is covered by all N toppings.