Probability and Statistics

This a problem set of a friend of mine.

1. A shipment of 12 calculators contains 3 defective calculators. In how many ways can a school purchase 5 of these calculators and receive at least two of the defective calculators? Draw the tree diagram.
2. A human gene carries a certain disease from mother to the child with a probability of 0.65. Suppose a mother of two children carries the gene. Also, assume that infection from child to child is independent. What is the probability that both children are infected with the disease? Draw the tree diagram.
3. A large industrial firm uses three different warehouses (A, B, and C) to store its manufactured product. From past records, it is known that 20% of the manufactured product are assigned to warehouse A, 50% are assigned to B, and 30% to warehouse C.
a. If it is known that 5% of the product in the warehouse A are defective, 4% in warehouse B are defective, and 8% in warehouse C are defective, what is the probability that if the product is selected at random from one of these warehouses that it will be defective?
b. What is the probability that if a defective product is selected at random that it came from warehouse C?