Question

(a) With conditional probability, ?(A|B), the axioms of probability hold for the event on the left side of the bar. A useful consequence is applying the complement rule to conditional probability. We have that ?(A|B) = 1 ? ?(A?|B). Prove this by showing that ?(A|B) + ?(A?|B) = 1 (Hint: just use the definition of conditional probability, a proof should be very short). (b) If two events A and B are independent, then we know ?(A ? B) = ?(A)?(B). A fact is that if A and B are independent, they so are all combinations of A, B?, ... etc. Show that if events A and B are independent, then ?(A? ? B?) = ?(A?)?(B?), and thus A? and B? are independent. (Hint: ?(A? ? B?) = 1 ? ?(A ? B). Then use addition rule and simplify.)

          (a) With conditional probability, ?(A|B), the axioms of probability hold for the event on the left side of the bar. A useful consequence is applying the complement rule to conditional probability. We have that ?(A|B) = 1 ? ?(A?|B).

Prove this by showing that ?(A|B) + ?(A?|B) = 1 (Hint: just use the definition of conditional probability, a proof should be very short).

(b) If two events A and B are independent, then we know ?(A ? B) = ?(A)?(B). A fact is that if A and B are independent, they so are all combinations of A, B?, ... etc.

Show that if events A and B are independent, then ?(A? ? B?) = ?(A?)?(B?), and thus A? and B? are independent. (Hint: ?(A? ? B?) = 1 ? ?(A ? B). Then use addition rule and simplify.)

(a) With conditional probability, ?(A|B), the axioms of probability hold for the event on the left side of the bar. A useful consequence is applying the complement rule to conditional probability. We have that ?(A|B) = 1 ? ?(A?|B).

Prove this by showing that ?(A|B) + ?(A?|B) = 1 (Hint: just use the definition of conditional probability, a proof should be very short).

(b) If two events A and B are independent, then we know ?(A ? B) = ?(A)?(B). A fact is that if A and B are independent, they so are all combinations of A, B?, ... etc.

Show that if events A and B are independent, then ?(A? ? B?) = ?(A?)?(B?), and thus A? and B? are independent. (Hint: ?(A? ? B?) = 1 ? ?(A ? B). Then use addition rule and simplify.)

Added by Scott C.

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