Recall when you condition on an event, that event becomes the new universe. Therefore the probability of A given B plus the probability of A' given B ought to be 1 according to the axioms of probability. In this problem, prove that P(A|B) + P(A'|B) = 1.

• (a) First, draw the sample space as a rectangle, and then include the sets A and B. Do not draw these sets so that they are disjoint. (Use the diagram on page 56 as a guide). Point to the region corresponding to the conditional event [A|B] and the conditional event [A'|B].
• (b) Prove P(A|B) + P(A'|B) = 1. DO NOT assume A and B are independent or disjoint. Model your proof as below: one step per line, with justification (which is provided).
  • P(A|B) + P(A'|B) = by the conditional probability formula, twice
  • = factor out the common quantity in each term = SAR in reverse, (not to A and B, but to E1 = A B and E2 = A0 B) = apply one of the properties of intersections & unions (which one do you need?) = use the fact that E' E = S, and then you're almost there!