Alex, Natasha, and Juan fill orders in a fast-food restaurant. Alex incorrectly fills 20% of the orders he takes. Natasha incorrectly fills 12% of the orders she takes. Juan incorrectly fills 5% of the orders he takes. Alex fills 30% of all orders, Natasha fills 45% of all orders, and Juan fills 25% of all orders. An order has just been filled.
a. What is the probability that Natasha filled the order?
b. If the order was filled by Juan, what is the probability that it was filled correctly?
c. Who filled the order is unknown, but the order was filled incorrectly. What are the revised probabilities that Alex, Natasha, or Juan filled the order?
d. Who filled the order is unknown, but the order was filled correctly. What are the revised probabilities that Alex, Natasha, or Juan filled the order?