Question

5% of all widgets are defective. there is a 30% chance that a widget was produced at the foo factory. if a widget was produced at the foo factory, then there is a 4% chance that the widget is defective.
a.) what is the probability that a widget was produced at the foo factory AND is defective?

b.) are "produced at the foo factory" and "defective" independent events? justify.

Answer 1

A.) P(defective | foo) = P(defective & foo)/P(foo)
  4% = P(defective & foo)/30% . . . . . . . . . plug in the given data
  0.04*0.30 = P(defective & foo) = 0.012 = 1.2%

The probability that a widget was produced at the foo factory and is defective is 1.2%.

b.) P(defective | foo) ≠ P(defective) (4% ≠ 5%), so the events P(defective) and P(foo) are NOT independent.

c.) P(foo | defective) = P(defective & foo)/P(defective)
  P(foo | defective) = 1.2%/5% = 24%

The probability that a widget was produced at the foo factory given it is defective is 24%.