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 72 71 T1 - Topping 1 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.
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 72 71 T1 - Topping 1 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.
Transcribed Image Text: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 TI
3/
T1 = Topping 1
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
